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Integration

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Indefinite Integral Suppose we know that a graph has gradient –2, what is the equation of the graph? There are many possible equations for graphs with gradient –2:

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Indefinite Integral Suppose now we have What are the possible equations for y what will give ?

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The process of obtaining the equation of the curve, given its derivative, is called Integration (or the antiderivative). Hence, Integration is the reverse process of differentiation.

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Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: Given:findWe don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant. Indefinite Integral

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In general, if,where c is a constant.

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Example 1: Integrate the following expressions with respect to x,where c is a constant.

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Indefinite Integral :, where c is an arbitrary constant and n 1. Some useful rules:

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Example 2: Find y in terms of x, of the following:,where c is a constant.

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If we have some more information we can find C. Given: and when, find the equation for. This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.

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Integration of Multiply by Outside first! Divide by Chain Rule: Apply when you have a composite function: f[g(x)] *n > 1 and coefficient of x ≠ 1

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, where c is a constant.

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Example 8: Integrate the following expressions with respect to x,where c is a constant.

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3.2 Definite Integrals Integration Symbol lower limit of integration upper limit of integration integrand variable of integration

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So far when integrating, there has always been a constant term left. Hence, such integrals are known as indefinite integrals. With definite integrals, we integrate a function between 2 points, and so we can find the precise value of the integral and there is no need for any unknown constant terms [the constant cancels out].integrating Definite Integrals Example:

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Example 9: EvaluateExpect a numerical answer!

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Example 10: If = 0, find the values of a.

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Example 11: Given that.Show that Hence evaluate Show Must make use of 1 st part result! From 1 st part, we know: Hence,

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Example 12: Given that.Show that Hence evaluate Show From 1 st part, we know: Hence,

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2. If the upper and lower limits are equal, then the integral is zero. 1. Reversing the limits changes the sign. 3. Constant multiples can be moved outside. Useful Rules: 4. Integrals can be added and subtracted. Using

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5. Intervals can be added (or subtracted.)

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Actual area under curve: Example 2:

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Area under a curve – Estimate using Trapezoid Rule Trapezoid Rule Divide curve into series of trapezoids (1,0) (2,6) (3,16) (4,30) (5,48) (6,70) (7,96)

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Area under a curve – Estimate using Simpson’s Rule (1,0) (3,16) (5,48) (7,96) (2,6) (4,30) (6,70)

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Area under Graphs – straight line x y y = f(x) = x x0 Area under the line y = x can be found using area of triangle Area =

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Let’s use another method to find the area under the line: x y y = f(x) = x 10 Area = We will use rectangles to approximate the area. Let’s start with n rectangles Hence, width of each rectangle =

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3.3 Integration by Substitution The variable of integration must match the variable in the expression. Don’t forget to substitute the value for u back into the problem! Example 1:

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One of the clues that we look for is if we can find a function and its derivative in the integrand. The derivative of is.Note that this only worked because of the 2x in the original. Many integrals can not be done by substitution. Example 2:

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Example 3: = = =

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Don’t forget to use the new limits. Example 4: = = when x = -1, u = 0 when x = 1, u = 2

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How can we find the area between these two curves? We could split the area into several sections, use subtraction and figure it out, but there is an easier way. Area under the curves

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Consider a very thin vertical strip. The length of the strip is: or Since the width of the strip is a very small change in x, we could call it dx.

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Since the strip is a long thin rectangle, the area of the strip is: If we add all the strips, we get:

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The formula for the area between curves is: We will use this so much, that you won’t need to “memorize” the formula!

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If we try vertical strips, we have to integrate in two parts: We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y.

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We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y. length of strip width of strip

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General Strategy for Area Between Curves: 1 Decide on vertical or horizontal strips. (Pick whichever is easier to write formulas for the length of the strip, and/or whichever will let you integrate fewer times.) Sketch the curves. 2 3 Write an expression for the area of the strip. (If the width is dx, the length must be in terms of x. If the width is dy, the length must be in terms of y. 4 Find the limits of integration. (If using dx, the limits are x values; if using dy, the limits are y values.) 5 Integrate to find area.

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