Presentation on theme: "Section 6.1 Volumes By Slicing and Rotation About an Axis"— Presentation transcript:
1 Section 6.1 Volumes By Slicing and Rotation About an Axis
2 Generalized CylinderA cylinder is a solid that is generated when a plane region is translated along a line or axis that is perpendicular to the region.If a cylindrical solid is generated y translating a region of area A through a distant h, then h is called the height of the cylinder, and the volume V of the cylinder is defined to beV=Ah =[area of a cross section] X [height]
4 Volumes By SlicingTo solve this problem, we begin by dividing the interval [a, b] into n subintervals, thereby dividing the solid into n slabs.If we assume that the width of the kth subinterval is xk, then the volume of the kth slab can be approximated by the volume A(xk)xk of a right cylinder of width (height) xk and cross-sectional area A(xk), where xk is a point in the kth subinterval.Adding these approximations yields the following Riemann sum that approximates the volume V:Taking the limit as n increases and the width of the subintervals approach zero yields the definite integral
5 Volume FormulaThe Volume of a solid can be obtained by integrating the cross-sectional area fromone end of the solid to the other.
6 How to Calculate the Volume To apply the formula in the definition to calculate the volume of a solid, take thefollowing steps:
7 ExampleA pyramid 3m high has a square base that is 3 m on a side. The cross section of the pyramid perpendicular to the altitude x m down from the vertex is a square x m on a side. Find the volume of the pyramid.
9 ExamplesExample: The region between the curve , 0 x 4, and the x-axis is revolved about the x-axis to generate a solid. Find its volume.
10 ExampleExample: Find the volume of the solid generated by revolving the region bounded by and the lines y=1, x=4 about the line y=1.
11 To find the volume of a solid generated by revolving a region between the y-axis and a curve x=R(y), c y d, about the y-axis, we use the same method with x replaced by y. In this case, the circular cross-section is A(y)= [radius]2 = [R(y)]2
12 ExampleExample: Find the volume of the solid generated by revolving the region between the y-axis and the curve x=2/y, 1 y 4, about the y-axis.
13 ExampleExample: Find the volume of the solid generated by revolving the region between the parabola x=y2+1 and the line x=3 about the line x=3.
16 Example: The region bounded by the curve y=x2+1 and the line y=-x+3 is revolved about the x-axis to generate a solid. Find the volume of the solid.
17 ExampleExample: The region bounded by the parabola y=x2 and the line y=2x in the first quadrant is revolved about the y-axis to generate a solid. Find the volume of the solid.
18 Section 6.2 Volumes by Cylindrical Shells The method of slicing in section 6.1 is sometimes awkward to apply. To overcomeThis difficulty, we use the same integral definition for volume, but obtain the area byslicing through the solid in a different way.
19 Volume of Cylindrical Shells A cylindrical shell is a solid enclosed by two concentric right circular cylinders.The volume V of a cylindrical shell with inner radius r1, outer radius r2, and heighth can be written asV=2 [1/2(r1+r2) ] h (r2-r1)So V=2 [ average radius ] [height] [thickness]
20 Method of Cylindrical Shells The idea is to divide the interval [a, b] into n subintervals, thereby subdividing the region R into n strips, R1, R2,, …, Rn.When the region R is revolved about the y-axis, these strips generate “tube-like” solids S1, S2, …, Sn that are nested one inside the other and together comprise the entire solid S.Thus the volume V of the solid can be obtained by adding together the volumes of the tubes; that isV=V(S1)+V(S2)+…+V(Sn).
22 Method of Cylindrical Shells Suppose that the kth strip extends from xk-1 to xk and that the width of the strip is xk. If we let xk* be the midpoint of the interval [xk-1, xk], and if we construct a rectangle of height f(xk*) over the interval, then revolving this rectangle about the y-axis produces a cylindrical shell of average radius xk*, height f(xk*), and thickness xk.Then the volume Vk of this cylindrical shell isVk=2xk*f(xk*) xkHence, we have
23 Volume by Cylindrical Shells about the y-axis Let f be continuous and nonnegative on [a, b] (0a<b), and let R be the region that is bounded above by y=f(x), below by the x-axis, and on the sides by the lines x=a and x=b. Then the volume V of the solid of revolution that is generated by revolving the region R about the y-axis is given byGenerally
25 ExampleExample: The region bounded by the curve , the x-axis, and the line x=4 is revolved about the y-axis to generate a solid. Find the volume of the solid.
26 ExampleSo far, we have used vertical axes of revolution. For horizontal axes, we replace the x’s with y’s.Example: The region bounded by the curve , the x-axis, and the line x=4 is revolved about the x-axis to generate a solid. Find the volume of the solid by the shell method.
27 ExamplesExample. Use cylindrical shells to find the volume of the solid generated when the region enclosed between , and the x-axis is revolved about the y-axis.Solution:
28 Arc Length Formula for Parametric Curves 6.3 Length of a Plane CurveArc Length Formula for Parametric Curves
35 Example Find the arc length of the curve from (1, 1) to Solution:
36 Dealing with Discontinuities in dy/dx At a point on a curve where dy/dx fails to exist, dx/dy may exist and we may be able toFind the curve’s length by expressing x as a function of y and applying the following:
37 ExampleExample: Find the length of the curve y=(x/2)2/3 from x=0 to x=2.