Step 1The region is bounded by the graphs of the equationsy sin x, y = 0, x=0, and xx$50For the representative rectangle, the radius of the solid of revolution isR(x)=sin xAccording to the disk method, the volume of the solid of revolution, when the area is revolved about the x axis is-=["[RE] =Therefore,V-sHence,v-=[" (Step 2Use the half angle identity for sin²x.sin²x1-cos 2xv=x["(ICV(sin(x))-=[x-sin 2x]Submit Skip (you cannot come back)Let1- cos 2xThe volume of the solid of revolution is VLetTherefore,- -COS XFor the above region,f(x)=sin xg(x)=02-0, D-11++So, the area of the entire region is7.Exercise (b)the centroid of the region bounded by the graphs of the equationsUse integration by parts.2Step 1The region is bounded by the graphs of the equationsy sin x, y=0, x=0, and x 8.AdThe area of the representative rectangle isdA= (sin.xTherefore,Using integration by partsyA.BXdx.Step 2The x-coordinate of the centroid for a region of constant density isx-1(*xx)-90x] x.[ udv-uv-[vou.X2dx.✓V*x-xsin x dxHOUX✓Differentiate with respect to x on both sides.du - dxdv - sin x dx.Integrate with respect to x on both sides.v-sin x dx>-(0)dxsin x dxcos(x)dx.✓xsin x dx-=[-x cos x + ["* cos x dx]"-X--x cos x + sin(x) IThe x-coordinate of the centroid of the region is

The region is bounded by the graphs of the equations y sin x, y = 0, x=0, and x-A. For the representative rectangle, the radius of the solid of revolution is R(x)-sin x According to the disk method, the volume of the solid of revolution, when the area is revolved about the x axis is v-=["[RO]~ V=. Therefore, Hence, 4 -=[" (sm) Stap 2 Use the half angle identity for sin²x. 1- cos2x sin x- V = * ✓ 2 1- cos 2x --x-in 2x Submit Skip (you cannot come back) HOU dx. The volume of the solid of revolution is V dx. yan(x) 6-9 K d dx >-(0)] Exercise () the centroid of the region bounded by the graphs of the equations Step 1 The region is bounded by the graphs of the equations y sin x, y=0, x=0, and x-x.

The region is bounded by the graphs of the equations y sin x, y = 0, x=0, and x-A. For the representative rectangle, the radius of the solid of revolution is R(x)-sin x According to the disk method, the volume of the solid of revolution, when the area is revolved about the x axis is v-=["[RO]~ V=. Therefore, Hence, 4 -=[" (sm) Stap 2 Use the half angle identity for sin²x. 1- cos2x sin x- V = * ✓ 2 1- cos 2x --x-in 2x Submit Skip (you cannot come back) HOU dx. The volume of the solid of revolution is V dx. yan(x) 6-9 K d dx >-(0)] Exercise () the centroid of the region bounded by the graphs of the equations Step 1 The region is bounded by the graphs of the equations y sin x, y=0, x=0, and x-x.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter9: Surfaces And Solids

Section9.3: Cylinders And Cones

Problem 6E: Suppose that r=12 cm and h=15 cm in the right circular cylinder. Find the exact and approximate a...

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