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32.7 Sketch and nd the area of: (a) the region to the left of the parabola x = 2y2 , to the right of the y-axis, and between y = 1 and y = 3; (b) the region above the line y = 3x 2, in the rst quadrant, and below the line y = 4; (c) the region between the curve y = x 3 and the lines y = x and y = 1. 32.8 Sketch the following regions and nd their areas: (a) The region between the curves y = x 2 and y = x 3. (b) The region between the parabola y = 4x 2 and the line y = 6x 2. (c) The region between the curves y = x, y = 1 and x = 4. (d) The region under the curve x + y = 1 and in the rst quadrant. (e) The region between the curves y = sin x, y = cos x, x = 0, and x = /4. (f) The region between the parabola x = y2 and the line y = x + 6.
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(g) The region between the parabola y = x 2 x 6 and the line y = 4. (h) The region between the curves y = x and y = x 3. (i) (j) The region in the rst quadrant between the curves 4y + 3x = 7 and y = x 2 . The region bounded by the parabolas y = x 2 and y = x 2 + 6x.
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(k) The region bounded by the parabola x = y2 + 2 and the line y = x 8. (l) The region bounded by the parabolas y = x 2 x and y = x x 2 .
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(m) The region in the rst quadrant bounded by the curves y = x 2 and y = x 4 . (n) The region between the curve y = x 3 and the lines y = x and y = 1.
APPLICATIONS OF INTEGRATION I: AREA AND ARC LENGTH
[CHAP. 32
32.9 Find the lengths of the following curves: (a) y = 1 x4 + 2 from x = 1 to x = 2. 8 4x (b) y = 3x 2 from x = 0 to x = 1. (c) y = x 2/3 from x = 1 to x = 8. (d) x 2/3 + y2/3 = 4 from x = 1 to x = 8. (e) y = 1 x5 + from x = 1 to x = 2. 15 4x 3 1 x(3 x) from x = 0 to x = 3. (f) y = 3 2 (1 + x 2 )3/2 from x = 0 to x = 3. 3
(g) 24xy = x 4 + 48 from x = 2 to x = 4. (h) y =
32.10 GC Use Simpson s rule with n = 10 to approximate the arc length of the curve y = f (x) on the given interval. (a) y = x 2 on [0, 1] (b) y = sin x on [0, ] (c) y = x 3 on [0, 5]
Applications of Integration II: Volume
The volumes of certain kinds of solids can be calculated by means of de nite integrals.
SOLIDS OF REVOLUTION
Disk and Ring Methods Let f be a continuous function such that f (x) 0 for a x b. Consider the region R under the graph of y = f (x), above the x-axis, and between x = a and x = b (see Fig. 33-1). If R is revolved about the x-axis, the resulting solid is called a solid of revolution. The generating regions R for some familiar solids of revolution are shown in Fig. 33-2.
Fig. 33-1 Theorem 33.1: The volume V of the solid of revolution obtained by revolving the region of Fig. 33-1 about the x-axis is given by V =
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