y 4 y 1 y Click here for answers. Click here for solutions. VOLUMES

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1 SECTION 7. VOLUMES 7. VOLUMES A Click here for answers. S Click here for solutions. 5 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.. y,, y ; about the -ais. y,, y ; about the -ais. y, y,, ; about the y-ais. y, y ; about the -ais 5. y, y,, ; about the y-ais 6 Find the volume of the solid obtained by rotating the region bounded by the given curves about the -ais. 6. y, y,, 7. y, y,,. y e, y,, 9. y s, y,,. y sec, y,,. y cos, y sin,,., y,, y. y,, 6, 5 Refer to the figure and find the volume generated by rotating the given region about the given line. y C, y yœ B, 6 Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line y s, y, 5; about the y-ais. y, y ; about y 7 9. y cos, y,, ; about y. y cos, y,, ; about y ; Use a graph to find approimate -coordinates of the points of intersection of the given curves. Then find approimately the volume of the solid obtained by rotating about the -ais the region bounded by these curves.. y, y s. y, y Sketch and find the volume of the solid obtained by rotating the region under the graph of f about the -ais... y ln, y, ; f if if if 5 5 Each integral represents the volume of a solid. Describe the solid. y 5. tan d 6. about the -ais f if if y y 6 dy y A, y 7. y y dy. y 6 d. about OA 5. about OC 6. about AB 7. about BC. about OA 9. about OC 9.. y y 5 5 d sin cos d Copyright, Cengage Learning. All rights reserved.. about BC. about AB. about OA. about OC. about BC 5. about AB. The base of S is the triangular region with vertices,,,, and,. Cross-sections perpendicular to the -ais are semicircles. Find the volume of S.

2 SECTION 7. VOLUMES 7. ANSWERS E Click here for eercises. S Click here for solutions V e ln ] d 7. V y y dy. V 6 9. V /. V / d cos cos d cos +cos d Solid obtained by rotating the region under the curve y tan, from to, about the -ais 6. Solid obtained by rotating the region bounded by the curve y and the lines y, y,and about the y-ais Solid obtained by rotating the region between the curves y and y about the y-ais. Solid obtained by rotating the region bounded by the curve y and the line y about the -ais 9. Solid obtained by rotating the region between the curves Copyright, Cengage Learning. All rights reserved e 9. ln. tan y 5 and y 5 about the -ais. Or: Solid obtained by rotating the region bounded by the curves y and y about the line y 5. Solid obtained by rotating the region bounded by the curves y +cosand y + sin and the line about the -ais.

3 SECTION 7. VOLUMES 7. SOLUTIONS E Click here for eercises.. V / d ] 5. V y ] dy y +y dy ] y/ y + y + +] 5 6. V + d + d + ] +. V y dy ydy y]. V + ] d d d ] 6. V d + d ] V / d /] +. V e d e d ] e e 9. The cross-sectional area is / + / +. Therefore, the volume is + d ln +] ln ln ln.. V sec d tan ] tan tan + ] tan. V / cos sin d / cos d sin ]/. V d + + d + d +] ]. V d + d + d + 5 d d Copyright, Cengage Learning. All rights reserved.. V d 6 ] 5. V y ] dy y 6 y] V y dy y y + 6 y] +

4 SECTION 7. VOLUMES Copyright, Cengage Learning. All rights reserved. 7. V ] d ]. V ] ] 5 5/ V y y ] dy 6 y y7] 7 7. V 6 d / 6 d 5 6y y 6 dy 5 ] d / / d ] + + / 5 5/ V y y ] dy 6y + y 6 +y 6y dy y + 7 y7 +y 6 y] + 7. V / ] d 5 5/. V y dy y7] 7 7. V d / + / d ] / + 5 5/ V y ] dy y y7] V e ln ] d 7. V 5 y + ] dy 6y y 6 dy 7 y y dy or 6,so V 6 6 ] ] d d 9. V / cos ] d / cos cos d. V / + cos ] d / cos +cos d. We see f rom the graph in Archived Problem 7..6 that the -coordinates of the points of intersection are.7 and., with +> on.7,.], sothe volume of revolution is about..7 + ] d..7 + d + 5 5] The -coordinates of the points of intersection are and.7,with > on,.7],sothevolumeof revolution is about.7 ] d ] d ] V d + d + 5 d V d + + d d ] ] The solid is obtained by rotating the region under the curve y tan, from to, about the -ais. 6. The solid is obtained by rotating the region bounded by the curve y and the lines y, y,and about the y-ais. 7. The solid is obtained by rotating the region between the curves y and y about the y-ais.

5 SECTION 7. VOLUMES 5. The solid is obtained by rotating the region bounded by the curve y and the line y about the -ais. 9. The solid is obtained by rotating the region between the curves y 5 and y 5about the -ais. Or: The solid is obtained by rotating the region bounded by the curves y and y about the line y 5.. The solid is obtained by rotating the region bounded by the curves y +cosand y + sin and the line about the -ais.. Since the area of a semicircle of diameter y is y,wehave V A d y d d ] d ] Copyright, Cengage Learning. All rights reserved.