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**10 Applications of Definite Integrals Case Study**

10.1 Finding Plane Areas by Integration 10.2 Volumes of Solids of Revolution Chapter Summary

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Case Study In some countries, there are very wide income gaps between the rich and the poor. To measure the inequality of income distribution in a society, we can make use of the Gini coefficient. The ‘Lorenz Curve’ is sketched such that any point (x, y) on the curve indicates that the poorest x% families share y% of the total income of all the families in the society. Yes, I know. How can we give a quantitative measure of the inequality of income distribution? If the Gini coefficient is close to 1, then there is a very wide income gap between the rich and the poor; if it is close to 0, then the income is uniformly distributed among the families.

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**10.1 Finding Plane Areas by Integration**

A. Area of the Region Bounded by a Curve and the x-axis In Chapter 9, we learnt that the definite integral of f (x) from a to b is defined as follows: As shown in the figure, if f (x) ³ 0 for a £ x £ b, then f (zi)Dx stands for the area of a very thin rectangle drawn under the curve y f (x). In this case, gives the area of the region bounded by the curve y f (x), the x-axis, the lines x a and x b. If f (x) ³ 0 in the interval a £ x £ b, then area of the shaded region

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**10.1 Finding Plane Areas by Integration**

A. Area of the Region Bounded by a Curve and the x-axis On the other hand, if f (x) £ 0 for a £ x £ b as shown in the figure, then f (zi)x is equal to the negative of the area of the thin rectangle. If f (x) £ 0 in the interval a £ x £ b, then area of the shaded region If f (x) takes both positive and negative values in the interval a £ x £ b, we can divide the shaded region into two parts: one part for f (x) ³ 0 and the other for f (x) £ 0. If f (x) ³ 0 in the interval a £ x £ c and f (x) £ 0 in the interval c £ x £ b, then area of the shaded region A1 + A2

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**Example 10.1T 10.1 Finding Plane Areas by Integration Solution:**

A. Area of the Region Bounded by a Curve and the x-axis Example 10.1T Find the area of the region bounded by the curve y x2, the axes and the line x 3. Solution: The required area

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**Example 10.2T 10.1 Finding Plane Areas by Integration Solution:**

A. Area of the Region Bounded by a Curve and the x-axis Example 10.2T Find the area of the region bounded by the curve y = x2 – 6x + 8, and the x-axis in the interval 0 £ x £ 4. Solution: When x2 – 6x + 8 = 0, The required area

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**10.1 Finding Plane Areas by Integration**

B. Area of the Region Bounded by y = f(x) and y = g(x) In the figure, y = f (x) and y = g(x) are two continuous functions such that f (x) ³ g(x) in the interval a £ x £ b. Once again, the area of the shaded region A can be approximated by a large number of thin, vertical rectangles of equal width ∆x. However, now the length of a rectangle is given by f (zi) – g(zi). where f (x) ³ g(x) in the interval a £ x £ b.

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**Example 10.3T 10.1 Finding Plane Areas by Integration Solution:**

B. Area of the Region Bounded by y = f(x) and y = g(x) Example 10.3T Find the area of the region bounded by the curves y = x2 and y = x3 . Solution: Consider the points of intersection of the two curves. (2) – (1): Hence the curves intersect when x = 0 and x = 1. The required area

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**Example 10.4T 10.1 Finding Plane Areas by Integration Solution:**

B. Area of the Region Bounded by y = f(x) and y = g(x) Example 10.4T Find the area of the region bounded by the curves y = x2, xy = 1 and the line y = 4x in the interval 0 £ x £ 1. Solution: Solving y = x2 and xy = 1, the point of intersection is (1, 1). Solving xy = 1 and y = 4x, the point of intersection is The required area

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**10.1 Finding Plane Areas by Integration**

B. Area of the Region Bounded by y = f(x) and y = g(x) In Example 10.3T, the function f (x) – g(x) does not change sign in the interval a £ x £ b, so the area of the region bounded by the two curves can be simply calculated by using the formula However, if y = f (x) and y = g(x) intersect in the interval as shown in the figure, we need to consider the shaded region on each side of the point of intersection independently.

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**Example 10.5T 10.1 Finding Plane Areas by Integration Solution:**

B. Area of the Region Bounded by y = f(x) and y = g(x) Example 10.5T Find the area of the region bounded by the curves and y = sin x in the interval 0 £ x £ p. Solution: At the point of intersection, The coordinates are On the left hand side of the intersection, is positive. On the right hand side of the intersection, is negative. The required area

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**10.1 Finding Plane Areas by Integration**

C. Area of the Region Bounded by a Curve and the y-axis Sometimes, the equation of a curve may be written in the form x = u(y), such as x = y2. Using similar arguments as we have studied in the previous section, we can find the area of the region bounded by the curve x = u(y), the y-axis and the lines y = c and y = d.

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**Example 10.6T 10.1 Finding Plane Areas by Integration Solution:**

C. Area of the Region Bounded by a Curve and the y-axis Example 10.6T Find the area of the region bounded by the curve y = x3 – 2, the y-axis and the line y = 6. Solution: For y = x3 – 2, The required area

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**Example 10.7T 10.1 Finding Plane Areas by Integration Solution:**

C. Area of the Region Bounded by a Curve and the y-axis Example 10.7T Find the area of the region bounded by the curve x = –y2 +2y + 3 and the y-axis in the interval –1 £ y £ 6. Solution: The required area

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**10.1 Finding Plane Areas by Integration**

D. Area of the Region Bounded by x = u(x) and x = v(y) As shown in the figure, x = u(y) and x = v(y) are two continuous functions such that u(y) ³ v(y) in the interval c £ y £ d. where u(y) ³ v(y) in the interval c £ y £ d.

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**Example 10.8T 10.1 Finding Plane Areas by Integration Solution:**

D. Area of the Region Bounded by x = u(x) and x = v(y) Example 10.8T In the figure, y 4x – 3 is a tangent to the curve y x4. Find the point of contact. Find the area of the region bounded by the curve y x4, the line y 4x – 3 and the x-axis. Solution: (a) Slope of y = 4x – 3 is 4. For y = x4, (b) The required area When 4x3 = 4, The point of contact is (1, 1).

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**10.2 Volumes of Solids of Revolution**

A. Solids of Revolution Revolved about the x-axis Consider the region bounded by the curve y , the x-axis, the y-axis and the line x = 4. If it is revolved about the x-axis, then it will generate a solid as shown in the figure. In this case, we call it a solid of revolution and the x-axis the axis of revolution. We will learn how to find the volume V of the solid of revolution using integration.

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**10.2 Volumes of Solids of Revolution**

A. Solids of Revolution Revolved about the x-axis Consider the curve of a non-negative function y = f (x) defined in the interval a £ x £ b. The area is approximated by n thin rectangles of width Dx = and length f (zi), where i = 1, 2, ... , n. If the first rectangle from the left is revolved about the x-axis to form a disc, the volume of the disc is If n is getting larger indefinitely, the width Dx tends to zero. Then the sum of the volumes of the discs will approach the actual volume of the solid of revolution. This method of finding the volume of the solid of revolution by integration is called the disc method.

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**10.2 Volumes of Solids of Revolution**

A. Solids of Revolution Revolved about the x-axis As shown in the figure, the shaded region enclosed by the line y = x (where r, h > 0) and x = h is revolved about the x-axis. Since the equation of the function is y = x with a = 0 and b = h, the required volume = We can also find the general formula for calculating the volume of a cylinder and a sphere using the disc method.

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**Example 10.9T 10.2 Volumes of Solids of Revolution Solution:**

A. Solids of Revolution Revolved about the x-axis Example 10.9T Find the volume of the solid generated by revolving the region enclosed by the curve y , the x-axis and the line x = 6 about the x-axis. Solution: The required volume

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**Example 10.10T 10.2 Volumes of Solids of Revolution Solution:**

A. Solids of Revolution Revolved about the x-axis Example 10.10T Find the volume of the solid generated by revolving the region enclosed by the curve y2 x2(x + 1) about the x-axis. Solution: When x2(x + 1) = 0, x = –1 or 0 The required volume

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**10.2 Volumes of Solids of Revolution**

A. Solids of Revolution Revolved about the x-axis Now, consider two continuous functions f (x) and g(x), where f (x) ³ g(x) > 0 in the interval a £ x £ b. If the region bounded by the curves y = f (x) and y = g(x) is revolved about the x-axis, then a hollow solid of revolution will be formed. The volume enclosed by the outer surface of this hollow solid is given by , while the volume of the empty space is equal to .

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**Example 10.11T 10.2 Volumes of Solids of Revolution Solution:**

A. Solids of Revolution Revolved about the x-axis Example 10.11T Find the volume of the solid generated when the region bounded by the curves y x2 + 2 and y –x2 + 4 is revolved about the x-axis. Solution: Substituting (1) into (2), The curves intersect at (1, 3) and (1, 3). The required volume

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**Example 10.12T 10.2 Volumes of Solids of Revolution Solution:**

A. Solids of Revolution Revolved about the x-axis Example 10.12T Find the volume of the solid generated when the region bounded by the curves y = ex and y = e–x in the interval 0 ≤ x ≤ ln 2 is revolved about the x-axis. Solution: The required volume

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**10.2 Volumes of Solids of Revolution**

B. Solids of Revolution Revolved about the y-axis Using the method, we can derive the formula for the volume of the solid generated by revolving about the y-axis. If u(y) ³ 0 in the interval c £ y £ d, then when the shaded region is revolved about the y-axis, the volume of the solid of revolution can be found by using the following formula:

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**Example 10.13T 10.2 Volumes of Solids of Revolution Solution:**

B. Solids of Revolution Revolved about the y-axis Example 10.13T Find the volume of the solid generated when the region between the y-axis and the curve x = tan y, where is revolved about the y-axis. Solution: The required volume

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**Example 10.14T 10.2 Volumes of Solids of Revolution Solution:**

B. Solids of Revolution Revolved about the y-axis Example 10.14T The region bounded by the curve y , the line 3x – 4y – 8 0 and the y-axis with x > 0 is revolved about the y-axis, find the volume of the solid generated Solution: The point of intersection is (4, 1). Substituting (1) into (2),

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**Example 10.14T 10.2 Volumes of Solids of Revolution Solution:**

B. Solids of Revolution Revolved about the y-axis Example 10.14T The region bounded by the curve y , the line 3x – 4y – 8 0 and the y-axis with x > 0 is revolved about the y-axis, find the volume of the solid generated Solution: The required volume

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**10.2 Volumes of Solids of Revolution**

B. Solids of Revolution Revolved about the y-axis Now let us consider two continuous functions u(y) and v(y), where u(y) ³ v(y) > 0 in the interval c £ y £ d. If the region bounded by the curves x = u(y) and x = v(y) is revolved about the y-axis, then the volume V of the hollow solid formed is given by:

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**Example 10.15T 10.2 Volumes of Solids of Revolution Solution:**

B. Solids of Revolution Revolved about the y-axis Example 10.15T A container is obtained by revolving the region bounded by the curves y 10 – x2, y 8 – x2 and the line y –2 about the y-axis. Find the volume of the material needed to make the container. Solution: Rewrite the equations of the curves y 10 – x2 and y 8 – x2 as x2 10 – y and x2 8 – y respectively. The required volume

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**Example 10.16T 10.2 Volumes of Solids of Revolution Solution:**

B. Solids of Revolution Revolved about the y-axis Example 10.16T Find the volume of the solid generated when the region bounded by the curve y 2 – (x + 1)2 and the line y 1 is revolved about the y-axis. Solution: Vertex (1, 2), The required volume Since the x-coordinates of the point on the left side of vertex less than –1, the equation of it is , where 1 £ y £ 2. The equation of the point on the right side of vertex is , where 1 £ y £ 2.

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**10.2 Volumes of Solids of Revolution**

C. Solids of Revolution Revolved about Lines Parallel to the Coordinate Axes Sometimes, the solid of revolution may be formed by revolving about a line parallel to the x-axis or the y-axis. In this case, we need to modify the formulas obtained in Sections 10.2 A and B in order to find the volume of the solid of revolution. For example, as shown in the figure, the region bounded by the curve y = f (x) and the line y = h (where h > 0) in the interval a £ x £ b is revolved about the line y = h to form the solid of revolution. If we translate the curve y = f (x) and the line y = h downwards by h units, then we can see that it is just the same as revolving the curve y = f (x) – h about the x-axis ! Thus, we can conclude that, the volume V of the solid of revolution is given by

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**Example 10.17T 10.2 Volumes of Solids of Revolution Solution:**

C. Solids of Revolution Revolved about Lines Parallel to the Coordinate Axes Example 10.17T Find the volume of the solid generated when the region bounded by the curve y , the lines y = 2 and x is revolved about the line y 2. Solution: when y 2, The required volume

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**Example 10.18T 10.2 Volumes of Solids of Revolution Solution:**

C. Solids of Revolution Revolved about Lines Parallel to the Coordinate Axes Example 10.18T Find the volume of the solid generated when the region bounded by the curve x cos y and the lines x y + 1 and x 2 in the interval is revolved about the line x 2. Solution: For x y + 1, when x 2, y 1 The required volume

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**10.2 Volumes of Solids of Revolution**

D. Shell Method In the disc method mentioned previously, the volumes of solids of revolution are found by approximating with circular discs. But in some situations, it may be easier to use the shell method, in which thin cylindrical shells are used to approximate the volume of the solid of revolution. For example, consider the region bounded by the curve y = f (x), the x-axis, the lines x = a and x = b as shown in the figure, where f (x) ³ 0 and 0 £ a £ b. A hollow solid is obtained when the region is revolved about the y-axis.

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**10.2 Volumes of Solids of Revolution**

D. Shell Method Now, if we approximate the region by a number of vertical rectangles and all these rectangles are revolved about the y-axis, each of them will generate a thin cylindrical shell of thickness Dx, height f (x) and radius x as shown in the figure. If Dx is very small, the volume of such a shell is approximately equal to the volume of a thin rectangular sheet of thickness Dx, height f (x) and length equal to the base circumference of the shell, that is, 2px. Hence the volume of a shell is approximately equal to 2px f (x)Dx and thus the total volume of the solid is

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**10.2 Volumes of Solids of Revolution**

D. Shell Method Similarly, if the region bounded by the function x = g(y), the lines y = c and y = d is revolved about the x-axis, then Note: 1. In the above discussion, f (x) and g(y) are assumed to be non-negative within the interval of integration. If not, the formulas should be modified as In the shell method, when the region is revolved about the y-axis, the volume is integrated along x; when it is revolved about the x-axis, the volume is integrated along y. It is opposite to the formulas we learnt in the disc method. Students should pay attention to this.

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**Example 10.19T 10.2 Volumes of Solids of Revolution Solution:**

D. Shell Method Example 10.19T Find the volume of the solid generated when the region bounded by the curve y , the x-axis and the line x 9 is revolved about y-axis. Solution: The required volume

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**Example 10.20T 10.2 Volumes of Solids of Revolution Solution:**

D. Shell Method Example 10.20T Find the volume of the solid generated when the region bounded by the curve y = x3, the x-axis and the line x = 1 is revolved about the line x = 2. Solution: The required volume

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**Chapter Summary 10.1 Finding Plane Areas by Integration**

1. For a £ b £ c, 2. For a £ b £ c, 3. For d £ e £ f, 4. For d £ e £ f,

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**Chapter Summary 10.2 Volumes of Solids of Revolution**

Hollow Solid of revolution about the x-axis In the figure, by the disc method, 2. Hollow Solid of revolution about the y-axis In the figure, by the disc method, When the region bounded by the curve y = f (x), the lines x = a, x = b and y = h is revolved about y = h, the volume of solid of revolution is given by

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**Chapter Summary 10.2 Volumes of Solids of Revolution**

In the figure, by the shell method, 1. 2. Similarly, when the region bounded by the curve x = g(y), the lines y = c and y = d is revolved about the x-axis,

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**Follow-up 10.1 10.1 Finding Plane Areas by Integration Solution:**

A. Area of the Region Bounded by a Curve and the x-axis Follow-up 10.1 The figure shows the curve y ln x. Find the area of the region bounded by the curve, the x-axis and the line x e. Solution: The required area

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**Follow-up 10.2 10.1 Finding Plane Areas by Integration Solution:**

A. Area of the Region Bounded by a Curve and the x-axis Follow-up 10.2 The figure shows the curve y x(x2 – 4x + 3). Find the area of the region bounded by the curve and the x-axis in the interval 0 £ x £ 3. Solution: The required area

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**Follow-up 10.3 10.1 Finding Plane Areas by Integration Solution:**

B. Area of the Region Bounded by y = f(x) and y = g(x) Follow-up 10.3 Find the area of the shaded region bounded by the curves y 9 – x2, y = 3x2 and the y-axis. Solution: Consider the points of intersection of the two curves. The required area Substituting (1) into (2), we have Hence the curves intersect when

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**Follow-up 10.4 10.1 Finding Plane Areas by Integration Solution:**

B. Area of the Region Bounded by y = f(x) and y = g(x) Follow-up 10.4 Find the area of the shaded region bounded by the curve y x2, the lines y 3x and y x + 2. Solution: Solving y x + 2 and y 3x, the point of intersection is (1, 3). Solving y x + 2 and y x2, the point of intersection is (2, 4). The required area

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**Follow-up 10.5 10.1 Finding Plane Areas by Integration Solution:**

B. Area of the Region Bounded by y = f(x) and y = g(x) Follow-up 10.5 Find the area of the region bounded by the curves y cos 2x and y sin x in the interval £ x £ Solution: At the points of intersection, The points of intersection are

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**Follow-up 10.5 10.1 Finding Plane Areas by Integration Solution:**

B. Area of the Region Bounded by y = f(x) and y = g(x) Follow-up 10.5 Find the area of the region bounded by the curves y cos 2x and y sin x in the interval £ x £ Solution: sin x cos 2x is positive. On the left hand side of On the right hand side of sin x cos 2x is negative. The required area

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**Follow-up 10.6 10.1 Finding Plane Areas by Integration Solution:**

C. Area of the Region Bounded by a Curve and the y-axis Follow-up 10.6 The figure shows the curve x ey. Find the area of the region bounded by the curve, the x-axis, the y-axis and the line y 2. Solution: The required area

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**Follow-up 10.7 10.1 Finding Plane Areas by Integration Solution:**

C. Area of the Region Bounded by a Curve and the y-axis Follow-up 10.7 The figure shows the curve x y3 – 3y2 – y + 3. Find the area of the region bounded by the curve and the y-axis in the interval −1 £ y £ 3 . Solution: The required area

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**Follow-up 10.8 10.1 Finding Plane Areas by Integration Solution:**

D. Area of the Region Bounded by x = u(x) and x = v(y) Follow-up 10.8 L is the tangent to the curve y x3 at the point (1, 1). (a) Find the slope of the tangent and hence find the equation of L. (b) Find the area of the region bounded by the curve y x3, the line L and the x-axis. Solution: (a) Slope of the tangent Equation of L: (b) For y = x3, For y = 3x – 2, The required area

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**Follow-up 10.9 10.2 Volumes of Solids of Revolution Solution:**

A. Solids of Revolution Revolved about the x-axis Follow-up 10.9 In the figure, find the volume of the solid generated by revolving the shaded region about x-axis. Solution: The required volume

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**Follow-up 10.10 10.2 Volumes of Solids of Revolution Solution:**

A. Solids of Revolution Revolved about the x-axis Follow-up 10.10 Find the volume of the solid generated by revolving the region enclosed by the curve y2 x(x – 1)2 about the x-axis. Solution: The required volume

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**Follow-up 10.11 10.2 Volumes of Solids of Revolution Solution:**

A. Solids of Revolution Revolved about the x-axis Follow-up 10.11 Find the volume of the solid generated when the region bounded by the curve y x2 + 1 and the line 2y 5 – x is revolved about the x-axis. Solution: The required volume Substituting (1) into (2), The curves intersect at and (1, 2).

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**Follow-up 10.12 10.2 Volumes of Solids of Revolution Solution:**

A. Solids of Revolution Revolved about the x-axis Follow-up 10.12 Find the volume of the solid generated when the region bounded by the curves y cos x and y sin x in the interval 0 £ x £ is revolved about the x-axis. Solution: The required volume

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**Follow-up 10.13 10.2 Volumes of Solids of Revolution Solution:**

B. Solids of Revolution Revolved about the y-axis Follow-up 10.13 Find the volume of the solid generated when the region bounded by the curve y , the line y 8 and the y-axis in quadrant I is revolved about the y-axis. Solution: The required volume

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**Follow-up 10.14 10.2 Volumes of Solids of Revolution Solution:**

B. Solids of Revolution Revolved about the y-axis Follow-up 10.14 The region bounded by the curve , the line y x – 3 and the y-axis with x > 0 is revolved about the y-axis. Find the volume of the solid generated. Solution: The point of intersection is (6, 3). Substituting (1) into (2), For y = x – 3, x = y + 3. The required volume

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**Follow-up 10.15 10.2 Volumes of Solids of Revolution Solution:**

B. Solids of Revolution Revolved about the y-axis Follow-up 10.15 A bowl is obtained by revolving the region bounded by the line y 3, the curves y x4 and y x4 – 1 about the y-axis. Find the volume of the material needed to make the bowl. Solution: Rewrite the equations of the curves y x4 and y x4 – 1 as and respectively. The required volume

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**Follow-up 10.16 10.2 Volumes of Solids of Revolution Solution:**

B. Solids of Revolution Revolved about the y-axis Follow-up 10.16 In the figure, the shaded region is bounded by the curve y (x – 1)2 – 1 and the x-axis. V is the vertex of the curve. (a) Write down the coordinates of V. (b) Find the volume of the solid of revolution generated by revolving the shaded region about the y-axis. Solution: (a) The equation of the right side of V is where –1 £ y £ 0. (b) The required volume Since the x-coordinates of the points on the left side of V, the equation of it is where –1 £ y £ 0.

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**Follow-up 10.17 10.2 Volumes of Solids of Revolution Solution:**

C. Solids of Revolution Revolved about Lines Parallel to the Coordinate Axes Follow-up 10.17 In the figure, the shaded region is bounded by the curve y cos x and the line y –1 in the interval –p £ x £ p. Find the volume of the solid of revolution generated by revolving the shaded region about the line y –1. Solution: The required volume

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**Follow-up 10.18 10.2 Volumes of Solids of Revolution Solution:**

C. Solids of Revolution Revolved about Lines Parallel to the Coordinate Axes Follow-up 10.18 In the figure, the shaded region is bounded by the curves x , x and the line x 2. Find the volume of The solid of revolution generated by revolving the shaded region about the line x 2. Solution: The required volume Substituting (1) into (2),

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**Follow-up 10.19 10.2 Volumes of Solids of Revolution Solution:**

D. Shell Method Follow-up 10.19 In the figure, the shaded region is bounded by the curve y (x – 1)2 – 1 and the x-axis. Using the shell method, find the volume of the solid of revolution generated by revolving the shaded region about the y-axis. Solution: When (x – 1)2 – 1 = 0 x – 1 – 1 or 1 x 0 or 2 The required volume

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**Follow-up 10.20 10.2 Volumes of Solids of Revolution Solution:**

D. Shell Method Follow-up 10.20 In the figure, the shaded region bounded by the curve x sin y and the y-axis in the interval 0 £ y £ p is revolved about the line y p. Find the volume of the solid of revolution generated. Solution: The required volume

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