Topics for Test #4 (Chap4, Exponential Integration, Chap5)

CHAPTER 4 INTEGRATIONThe material in this chapter is the climax of a first-semester calculus course. Riemann sums are used to approximate the area under a curve, and the definite integral is defined as the limit of Riemann sums. The marriage of limits, derivatives and integration is expressed in the Fundamental Theorem of Calculus, the reward of a semester of hard work.

4.1 Areas and DistancesOverviewThe area of a region bounded by the graph of a function over some interval (more simply, the area under a curve) is approximated by summing the areas of rectangles whose heights are given by the values of the function (that is, by evaluating Riemann sums). In this section, regions are assumed to be bounded by the graph of a positive function.

Lecture

Study the Area Problem section and Watch the Lecture Video on page 284. Do Examples 1 . Skip page 286. Study the Area definition, and do the Wolfram Demonstration on page 289.

Familiarize with the Sigma notation (watch the Lecture Video ), and do Example 3 on page 290. Study The Distance Problem and Example 4 .

Homework:Do the assigned exercises in WebAssign.

4.2 Definite IntegralsOverviewA good deal of information is presented in this section. The definite integral is defined as the limit of Riemann sums (with a general partition), and properties of definite integrals are explored. Regions above the x-axis make positive contributions to a Riemann sum, while regions below the x-axis make negative contributions. Finally, we present methods for evaluating definite integrals.LectureWe approximate the area under the graph of a function with sums of rectangles whose heights are given by values of the function. These sums generally approach a limit as the number of rectangles increases. See the definition of Definite Integrals. Skip the videos on page 297, do Example1, skip pages 299-300, and do Example 4 .

For additional videos, watch the video Approximating areas Using Rectangles at

http://www.wonderhowto.com/how-to-approximate-area-under-curve-using-rectangles-341505/, a Right Riemann Sum at http://www.youtube.com/watch?v=uoipFxrjge4&NR=1 and the first 7.5 minutes of a left and Right Riemann Sums athttp://www.youtube.com/watch?v=C58TZEzYSRs&feature=mfu_in_order&playnext=1&videos=O5QxfxXvNNE

Study the Midpoint Rule, and watch the Lecture Video , do Example 5 and the Tool for Enriching Calculus on page 302

Study the Properties of the Definite Integral and watch the Lecture Video on page 303. Do Examples 6, 7, 8

For an additional video, watch the video (first 6 minutes) of a Midpoint Riemann Sum athttp://www.youtube.com/watch?v=gQqm4BkFqfw&feature=mfu_in_order&playnext=1&videos=BOvau__CTe8

You can find Riemann Sums applets at http://www.math.tamu.edu/AppliedCalc/Classes/Riemann/index.html or athttp://www.scottsarra.org/applets/calculus/RiemannSums.html

For additional videos, watch video of How to define an Indefinite Integral at http://www.wonderhowto.com/how-to-define-indefinite-integral-calculus-341498/And The Indefinite Integral-Understanding the Definition at http://www.youtube.com/watch?v=LkdodHMcBuc&feature=related

4.3 Fundamental Theorem of CalculusOverviewA discussion of area functions and their properties leads to the Fundamental Theorem of Calculus, where we discover the inverse relationship between differentiation and integration. The FTC also gives an efficient method for evaluating definite integrals.Lecture

Start with the area function A(x) =∫a

x

f ( t ) dt ,do Example 1 and study the Tutorial on page

311. Look at the Fundamental Theorem of Calculus Part 1(FTC1). FTC1 says that the derivative of a definite integral with respect to the upper limit is the integrand evaluated at the upper limit. FTC1 takes the derivative of an integral. Skip the proof of FTC1. Do Examples 2 , 4.Watch the video on The Fundamental Theorem of Calculus (Part 1) at http://www.youtube.com/watch?v=PGmVvIglZx8

Look at the Fundamental Theorem of Calculus Part 2(FTC2) on page 315.

Watch the Lecture Video on page 312, and do Examples 5 , 6, 7, 8

You can watch the videos on The Fundamental Theorem of Calculus (Part 2)

http://www.brightstorm.com/math/calculus/the-definite-integral/the-fundamental-theorem-of-calculus

Notice that the FTC links two of the most important processes in calculus, and that differentiation and integration “undo” one another (the Differentiation and Integration as a

Reverse Process section, and Watch the Lecture Video on page 317.).

Homework:Do the assigned exercises in WebAssign. In the homework section in BB, you will find (highlighted) problems not in WebAssign that are part of the required homework. These problems will not be collected but may appear in tests.Note: All Wolfram Demonstrations are optional, but highly recommended.

4.4 Indefinite Integrals and the Net Change Theorem.OverviewIn this section, we introduce a notation for antiderivatives and use them to evaluate Definite Integrals and indefinite Integrals

Lecture Study the concept the indefinite integrals and memorize the Table of Indefinite Integrals. Watch

the Lecture Video on page 322. Do Examples 1, 2 , 3, 4, 5 and watch the Lecture Video

on page 323.Look at the different applications of the Net Change Theorem. Do Example 6 . Make sure you understand the difference between displacement and total distance traveled. The displacement is Net Area of the velocity function, and the velocity is the Total Area of the velocity function (area above the horizontal axis minus area below the horizontal axis)For an extra video, go to http://www.youtube.com/watch?v=Lb8QrUN6Nck&feature=fvw

Homework:Do the assigned exercises in WebAssign. In the homework section in BB, you will find (highlighted) problems not in WebAssign that are part of the required homework. These problems will not be collected but may appear in tests.Note: All Wolfram Demonstrations are optional, but highly recommended.

4.5 The Substitution RuleOverview

Substitution (or change of variables) is one of the most important analytical techniques for evaluating integrals.

Lecture The most effective way to learn the method of substitution is to do as many problems as possible. You need a lot of practice to see the important patterns. Begin with the Substitution rule. Do the

Tutorial on page 330 and the Lecture Video on page 331. Do the indefinite integrals Examples 1, 2, 3 , 4, 5. Continue with the Substitution rule for definite integrals, Examples

6,7 and the Lecture Video on page 334. Study the Integrals of symmetric Functions and do Examples 8 , 9.

Homework:Do the assigned exercises in WebAssign. In the homework section in BB, you will find (highlighted) problems not in WebAssign that are part of the required homework. These problems will not be collected but may appear in tests.Note: All Wolfram Demonstrations are optional, but highly recommended.

Exponential Integration

Evaluate the following integrals. Use the chain rule if needed.

1.

2. ∫5 e2 x √1+e2 x dx

3. ∫ 5

e−6 xdx

4. ∫ e tan (2 x )sec2 (2 x )dx

5. ∫ e√ x

√xdx

6. ∫ ep2dx

7. , where a and b constant

8.

Answers:

1.

−43

cos (e3 x) +C, 2.

53 √(e2 x+1 )3+C

, 3.

56

e6 x+C, 4.

12

e tan (2 x )+C, 5. 2 e√x+C , 6. e p2

x+C ,

7.

ab

ebx+C, 8.

12

(e x+1 )2+C

CHAPTER 5 APPLICATIONS OF INTEGRATIONThis chapter is devoted to using the definite integral as a problem-solving tool in a variety of settings. We investigate the net change of a quantity whose rate of change is known and develop area and volume formulas.

5.1 Regions Between CurvesOverviewThe method for finding the area of a region under a single curve (Chapter 4) is generalized so that areas bounded by two or more curves can be computed.Lecture Begin by looking at the definition for the area of a region between two curves that intersect no

more than twice over some interval. Watch the Lecture Video on page 344 and do Example 1 . Remember that a sketch of the region in question is indispensable for determining the limits of integration and the order of subtraction in the integrand. Plotting 3 or 4 points is often sufficient for area sketches; we usually don’t need a detailed graph to visualize the region. It’s a good idea to draw a typical rectangle from the Riemann sum, particularly when integration is with respect to y. In many occasions, to find the limits of integration, you need to set the equations of the bounding

curves equal to one another to obtain the points of intersections. Watch the Lecture Video on page 346 and do Example 2.In contrast to the idea of net area in Chapter 4, it doesn’t matter if the bounding curves drop below the x-axis. As long as g(x) is subtracted from f (x), where g(x) < f (x), the height of a typical rectangle from the Riemann sum will be positive.In cases where the curves intersect more than twice, you will have to split the region.

Watch the two Lectures Videos on page 347 and do Example 5 . Sometimes if the area yields integrals that are not easily to integrate on the x-axis, you need to integrate with respect to y. There are two reasons to change from integration on the x-axis to integration on the y-axis. You may be able compute an area with fewer integrals, or integration with respect to y may be easier (for instance, a polynomial versus a root function). Do Example 6 Homework:Do the assigned exercises in WebAssign. In the homework section in BB, you will find (highlighted) problems not in WebAssign that are part of the required homework. These problems will not be collected but may appear in tests.Note: All Wolfram Demonstrations are optional, but highly recommended.

5.2 Volume by SlicingOverviewVolumes of solids whose cross-sectional areas are described with a known function are computed by employing the slice-and-sum method. This method is then applied to find a formula for the volume of a solid of revolution using disks and washers.Lecture Study the Disk Method. Look at the formula for the Disk Method About the x-Axis, and do

Example 2 . Look at the formula for the Disk Method About the y-Axis, and do Example 3 .

Study the Washer Method. Look at the formula for the Washer Method About the x-Axis, and do Example 4. See the Wolfram Demonstration , the Tools for enriching Calculus , and the two Lectures

Videos at the top of page 356.

Use the Washer Method about the any horizontal or vertical line, and do Examples 5 and 6.

Watch the two Lectures Videos at the bottom of page 356 and the Lectures Video on page 358. Skip Examples 7,8,9.

Notice that the Disk Method is a special case of the Washer Method.Adopt the habit of drawing little Riemann sum rectangles (or simply vertical or horizontal lines) to make the connection between the figure and the corresponding integral formula. Better still is if you can rotate the line around the axis of revolution and sketch the resulting cross section.

Homework:Do the assigned exercises in WebAssign. In the homework section in BB, you will find (highlighted) problems not in WebAssign that are part of the required homework. These problems will not be collected but may appear in tests.

5.3 Volume by Cylindrical ShellsOverviewVolumes of solids of revolution are computed with the shell method.

Lecture The biggest challenge with this section is minimizing confusion. Students must keep track of several solids of revolution formulas: In their minds, there are 8 of them (disks and washers about the x- and y-axis, and shells with one function and two about each axis). It doesn’t help that the variable of integration changes from disks to shells—that is, a region revolved about the x-axis using the disk method requires integration with respect to x, while the volume of the same solid using shells requires integration with respect to y. Study the Shell Method on . Look at the formula for Volume by the Shell Method, watch the

Lectures Video , and do Examples 1 , 2 and 3 on page 365.

With the disk/washer method, the variable of integration matches the axis of rotation (for example, using the disk/washer method for a region rotated about the x-axis requires integration with respect to x), but with the shell method, the variable of integration is different than the axis of rotation (using the shell method about the x-axis requires integration with respect to y).In general, you can use either disks or shells to find volumes. One integral will be easier to compute than the other.

Homework:Do the assigned exercises in WebAssign.Chapter 4 Key Terms and ConceptsLeft/right/midpoint Riemann sum Net area Definite integral Properties of definite integrals Fundamental Theorem of Calculus, Parts 1 and 2 Integrals of symmetric functions (Theorem 5.4) Substitution Rule (Change of Variables) for Indefinite Integrals Substitution Rule (Change of Variables) for Definite Integrals Exponential IntegrationChapter 5 (sections 5.1-5.3) Key Terms and ConceptsArea of a region between two curves with respect to x and y Volume by the disk/washer methodVolume by the shell method