CALCULUS II Chapter 5

Definite Integral

Example

Properties of the Definite Integral 1: 2: 3: 4: 5: 6:

7:

8: 9: 10: 11:

12:

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 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.

Table of Indefinite Integrals

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

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.

Visualization

Fundamental Theorem of Calculus (part 2)

The Fundamental Theorem of Calculus Ex.

The Fundamental Theorem of Calculus Ex.

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

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. It does not matter what constant we use! (Limits are reversed.) (Chain rule is used.) We split the integral into two parts.

More Ex’s on the FTC

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 Pick u, compute du Sub in Integrate

Ex. Evaluate

Ex. Evaluate

Examples

Shortcuts: Integrals of Expressions Involving ax + b Rule

Evaluating the Definite Integral Ex. Calculate

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

Examples

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

Net or Total Change as the Integral of a Rate Integral of a rate of change Total change over

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? 200ft 100ft 700 feet

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

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.

velocity graph position graph Displacement: Distance Traveled:

Examples A particle moves along a line so that its velocity at time t is (in meters per second): Find the displacement of the particle during the period Find the total distance travelled during the same period

Examples A factory produces bicycles at a rate of (in t weeks) How many bicycles were produced from day 8 to 21?

Examples At 7 AM, water begins leaking from a tank at a rate of (t is the number of hours after 7 AM) How much water is lost between 9 and 11 AM?

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.

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

69 Review Recall derivatives of inverse trig functions

70 Integrals Using Same Relationships When given integral problems, look for these patterns

71 Identifying Patterns For each of the integrals below, which inverse trig function is involved? Hint: use completing the square

72 Warning Many integrals look like the inverse trig forms Which of the following are of the inverse trig forms? If they are not, how are they integrated?