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

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

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

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.

Power Rule for the Indefinite Integral, Part I Ex.

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

Sum and Difference Rules Ex. Constant Multiple Rule Ex.

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

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

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

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

Ex. Evaluate

Shortcuts: Integrals of Expressions Involving ax + b Rule

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

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

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

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.”

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.)

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

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

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

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

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

The Fundamental Theorem of Calculus Ex.

Evaluating the Definite Integral Ex. Calculate

Substitution for Definite Integrals Ex. Calculate Notice limits change

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

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. 

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. 