 # CALCULUS II Chapter 5.

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CALCULUS II Chapter 5

Definite Integral

Example

Properties of the Definite Integral
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Indefinite Integrals or Antiderivatives
You should distinguish carefully between definite and indefinite integrals. A definite integral is a number, whereas an indefinite integral is a function (or family of functions).

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

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

Constant of Integration
Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant. Notice 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 ex and bx

Sum and Difference Rules
Ex. Constant Multiple Rule Ex.

Table of Indefinite Integrals

Fundamental Theorem of Calculus (part 1)
If is continuous for , then

Visualization

Fundamental Theorem of Calculus (part 2)
1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

The Fundamental Theorem of Calculus
Ex.

First Fundamental Theorem:
1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

1. Derivative of an integral.
2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

The upper limit of integration does not match the derivative, but we could use the chain rule.

The lower limit of integration is not a constant, but the upper limit is.
We can change the sign of the integral and reverse the limits.

Neither limit of integration is a constant.
We split the integral into two parts. It does not matter what constant we use! (Limits are reversed.) (Chain rule is used.)

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 get Integrate Back Substitute

Ex. Evaluate Pick u, compute du Sub in Integrate Sub in

Ex. Evaluate

Ex. Evaluate

Shortcuts: Integrals of Expressions Involving ax + b
Rule

Evaluating the Definite Integral
Ex. Calculate

Computing Area Gives the area since 2x3 is nonnegative on [0, 2].
Ex. Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and the graph of Gives the area since 2x3 is nonnegative on [0, 2]. Antiderivative Fund. Thm. of Calculus

Substitution for Definite Integrals
Ex. Calculate Notice limits change

The Definite Integral As a Total
If r(x) is the rate of change of a quantity Q (in units of Q per unit of x), then the total or accumulated change of the quantity as x changes from a to b is given by

The Definite Integral As a Total
Ex. If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by

A honey bee makes several trips from the hive to a flower garden.
The velocity graph is shown below. What is the total distance traveled by the bee? 700 feet 200ft 200ft 200ft 100ft

What is the displacement of the bee?
100 feet towards the hive 200ft 200ft -200ft -100ft

To find the displacement (position shift) from the velocity function, we just integrate the function. The negative areas below the x-axis subtract from the total displacement. To find distance traveled we have to use absolute value. Find the roots of the velocity equation and integrate in pieces, just like when we found the area between a curve and the x-axis. (Take the absolute value of each integral.) Or you can use your calculator to integrate the absolute value of the velocity function.

Displacement: Distance Traveled: velocity graph position graph Every AP exam I have seen has had at least one problem requiring students to interpret velocity and position graphs.

In the linear motion equation:
V(t) is a function of time. For a very small change in time, V(t) can be considered a constant. We add up all the small changes in S to get the total distance.

We add up all the small changes in S to get the total distance.
As the number of subintervals becomes infinitely large (and the width becomes infinitely small), we have integration.

This same technique is used in many different real-life problems.

Example 5: National Potato Consumption The rate of potato consumption for a particular country was: where t is the number of years since 1970 and C is in millions of bushels per year. For a small , the rate of consumption is constant. The amount consumed during that short time is

Example 5: National Potato Consumption The amount consumed during that short time is We add up all these small amounts to get the total consumption: From the beginning of 1972 to the end of 1973: million bushels