121616949-math.206 - s Its cross-sections perpendicular to...

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192 Chapter 9 Applications of Integration 7. Find the volume of the solid obtained by revolving the region bounded by y = sin x , the y -axis, and the lines y = 1 and x = π/ 2 around the x -axis. 8. Let S be the region of the xy -plane bounded above by the curve x 3 y = 64, below by the line y = 1, on the left by the line x = 2, and on the right by the line x = 4. Find the volume of the solid obtained by rotating S around (a) the x -axis, (b) the line y = 1, (c) the y -axis, (d) the line x = 2. 9. The equation x 2 / 9 + y 2 / 4 = 1 describes an ellipse. Find the volume of the solid obtained by rotating the ellipse around the x -axis and also around the y -axis. These solids are called ellipsoids ; one is vaguely rugby-ball shaped, one is sort of flying-saucer shaped, or perhaps squished-beach-ball-shaped. Figure 9.12 Ellipsoids. ( JA ) 10. Use integration to compute the volume of a sphere of radius r . You should of course get the well-known formula 4 πr 3 / 3. 11. A hemispheric bowl of radius r contains water to a depth h . Find the volume of water in the bowl. 12. The base of a tetrahedron (a triangular pyramid) of height
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Unformatted text preview: s . Its cross-sections perpendicular to an altitude are equilateral triangles. Express its volume V as an integral, and find a formula for V in terms of h and s . Verify that your answer is (1 / 3)(area of base)(height). 13. The base of a solid is the region between f ( x ) = cos x and g ( x ) =-cos x ,-π/ 2 ≤ x ≤ π/ 2, and its cross-sections perpendicular to the x-axis are squares. Find the volume of the solid. ⇒ 9.4 Average value of a function The average of some finite set of values is a familiar concept. If, for example, the class scores on a quiz are 10, 9, 10, 8, 7, 5, 7, 6, 3, 2, 7, 8, then the average score is the sum of these numbers divided by the size of the class: average score = 10 + 9 + 10 + 8 + 7 + 5 + 7 + 6 + 3 + 2 + 7 + 8 12 = 82 12 ≈ 6 . 83 . Suppose that between t = 0 and t = 1 the speed of an object is sin( πt ). What is the average speed of the object over that time? The question sounds as if it must make sense,...
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