AB Student Notes: Area and Volume

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Eustace Carter
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1 AB Student Notes: Area and Volume An area and volume problem has appeared on every one of the free response sections of the AP Calculus exam AB since year 1. They are straightforward and only occasionally have anything out of the ordinary. Most often, these problems will appear on the non-calculator free response section. This is because there are calculator programs which will give the set up which students could then copy onto their answer paper. You may be asked to find the point(s) of intersection of the given functions in order to use them as the limits of integration. Show the equation you are solving, not just the solution. Finding the values is not enough: you do not earn the point until you use the value correctly as a limit of integration somewhere in the problem. Simple questions may also appear on the multiple-choice section. For example, if volume by crosssections has not appeared in a free response, it will likely appear on the multiple choice. The topics of area, volume by cross section and volumes of revolution do show up on the exam, either on the multiple choice or on the free response. What should you know how to do? Solve equations. Remember to show the equation you are solving. Graph functions. Often the graph is given; occasionally you will want to sketch your own graph to be sure which curve is on top. Find the area of a region enclosed by two functions over an interval. You may have to find the endpoints of the interval to use as limits of integration. Find the volume when the region is revolved around an axis or a line using the washer or disk method. If you learned the method of cylindrical shells, you may use it; however, any volume you are asked for can be done by the washer/disk method. Find volumes of solid figures with regular cross sections squares, equilateral triangles, semicircles, etc. Though volume formulas for related rates problems that involve cones or cylinders are generally provided, area formulas are not provided. Be sure to know the basic area formulas. Rounding and Simplifying Some parts may ask you to, Write, but do not evaluate, an integral expression. In other words you are to set up the integral for the area or volume, but you do not have to find an antiderivative or the numerical answer. If you are asked to find the numerical answer, remember that answers need not be simplified either numerically, so DO NOT SIMPLIFY. If you get leave it! A correct answer simplified incorrectly will lose the point. Do not try to change your answer to a decimal. If you are asked to find the point(s) of intersection of the given functions in order to use them as the limits of integration, show the equation you are solving, not just the solution. You may use your calculator to find the values.

2 You are expected to use your calculator to evaluate the definite integrals you set up. Remember to show the integral with limits of integration you are evaluating in standard notation. On the calculator allowed section you do not need to show an antiderivative (and an incorrect antiderivative could cost you a point). The test directions say, Your final answers must be accurate to three places after the decimal point. This is what accurate to three places after the decimal point means: the first three digits after the decimal point must be correct by truncating or rounding. However, you are not required to round or truncate to three places. Anything written after the third decimal place, right or wrong, is ignored. Thus any of these values for π (which should be left asπ ) is acceptable: , 3.141, 3.14, , , Once again: DON T ROUND, a correct answer that is then rounded incorrectly. Page

5 Multiple Choice. Non-Calculator. Choose the best answer. 1. What is the area of the region between the graphs of y x and y x from x = 0 to x =? A. 3 B. 8 3 C. 4 D E If the region enclosed by the y-axis, the line y =, and the curve y y-axis, the volume of the solid generated is x is revolved about the A. 3 5 B C D. 8 3 E. 3. The base of a solid is the region in the first quadrant bound by the graph of y x and the coordinate axes. If every cross section of the solid perpendicular to the y-axis is a square, the volume of the solid is given by A. y dy B. 0 D. x dx E. 0 y dy C. x dx 0 0 x dx 0 4. The area of the region enclosed by the graph of y x 1 and the line y = 5 is A B C. 8 3 D. 3 3 E. 8 Calculator. Choose the best answer Let R be the region enclosed by the graph of y 1 ln cos x x and x. The closest integer approximation of the area of R is 3 3, the x-axis, and the lines A. 0 B. 1 C. D. 3 E. 4 Page 5

6 6. The region bounded by the graph of y x x and the x-axis is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an equilateral triangle. What is the volume of the solid? A B C D E The base of a solid is in a region in the first quadrant bounded by the x-axis, the y-axis and the line x + y = 8 as shown in the figure above. If cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid? A B C D E If 0 k and the area under the curve y cos x from x k to x is 0.1, then k = A B C D E What is the area of the region in the first quadrant enclosed by the graphs of y cos x, y x and the y-axis? A B C D E Page 6

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10 AB Student Notes: Area and Volume Multiple Choice: 1D, A, 3B, 4D, 5B, 6D, 7C, 8D, 9C, 10D Page 10

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CHAPTER 6: APPLICATIONS OF INTEGRALS

(Exercises for Section 6.1: Area) E.6.1 CHAPTER 6: APPLICATIONS OF INTEGRALS SECTION 6.1: AREA 1) For parts a) and b) below, in the usual xy-plane i) Sketch the region R bounded by the graphs of the given