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The Integral chapter 5 The Indefinite Integral Substitution The Definite Integral As a Sum The Definite Integral As Area The Definite Integral: The Fundamental Theorem of Calculus

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Antiderivative An antiderivative of a function f is a function F such that Ex.An antiderivative of since is

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means to find the set of all antiderivatives of f. The expression: read “the indefinite integral of f with respect to x,” Integral sign Integrand Indefinite Integral x is called the variable of integration

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Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant. Notice Constant of Integration Represents every possible antiderivative of 6x.

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Power Rule for the Indefinite Integral, Part I Ex.

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Power Rule for the Indefinite Integral, Part II Indefinite Integral of e x and b x

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Sum and Difference Rules Ex. Constant Multiple Rule Ex.

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Integral Example/Different Variable Ex. Find the indefinite integral:

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Position, Velocity, and Acceleration Derivative Form If s = s(t) is the position function of an object at time t, then Velocity = v =Acceleration = a = Integral Form

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Integration by Substitution Method of integration related to chain rule differentiation. If u is a function of x, then we can use the formula

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Integration by Substitution Ex. Consider the integral: Sub to getIntegrateBack Substitute

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Ex. Evaluate

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Shortcuts: Integrals of Expressions Involving ax + b Rule

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Riemann Sum If f is a continuous function, then the left Riemann sum with n equal subdivisions for f over the interval [a, b] is defined to be

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Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable) Note that the integral does not depend on the choice of variable. If f is a function defined on [a, b], the definite integral of f from a to b is the number provided that this limit exists. If it does exist, we say that f is integrable on [a, b]. Definition of a Definite Integral

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Approximating the Definite Integral Ex. Calculate the Riemann sum for the integral using n = 10.

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The Definite Integral is read “the integral, from a to b of f(x)dx.” Also note that the variable x is a “dummy variable.”

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Area Under a Graph a b Idea: To find the exact area under the graph of a function. Method: Use an infinite number of rectangles of equal width and compute their area with a limit. Width: (n rect.)

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Approximating Area Approximate the area under the graph of using n = 4.

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Area Under a Graph a b f continuous, nonnegative on [a, b]. The area is

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Geometric Interpretation (All Functions) Area of R 1 – Area of R 2 + Area of R 3 a b R1R1 R2R2 R3R3

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Area Using Geometry Ex. Use geometry to compute the integral Area = 2 Area =4

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Fundamental Theorem of Calculus Let f be a continuous function on [a, b]. 2. If F is any continuous antiderivative of f and is defined on [a, b], then

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The Fundamental Theorem of Calculus Ex.

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Evaluating the Definite Integral Ex. Calculate

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Substitution for Definite Integrals Ex. Calculate Notice limits change

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Computing Area Ex. Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and the graph of Gives the area since 2x 3 is nonnegative on [0, 2]. AntiderivativeFund. Thm. of Calculus

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Quiz Find the area between the x-axis and the curve from to. On the TI-89: If you use the absolute value function, you don’t need to find the roots. pos. neg.

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Quiz Find the area between the x-axis and the curve from to. On the TI-89: If you use the absolute value function, you don’t need to find the roots. pos. neg.

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