Use cylindrical shells to compute the volume of the solid formed by revolving the region bounded by about y = 10.

Question

Use cylindrical shells to compute the volume of the solid formed by revolving the region bounded by  about y = 10.

Quizplus · Accepted Answer

To use cylindrical shells, we want to integrate along the axis of rotation (in this case, the y-axis). We can split the region bounded by 11eaa948_cd04_6c30_84bc_dd8e5ddf7903_TB2342_11 into infinitesimal strips of width "dy". Each strip has a height of (y-10) because it is the distance between the y-coordinate and the axis of rotation. The radius of each strip will be the distance from the y-axis to the right boundary of the region, which is 11eaa948_cd04_6c30_84bc_dd8e5ddf7903_TB2342_11 - y^2. Thus, the volume of each shell will be:

V = 2π(y-10)(11eaa948_cd04_6c30_84bc_dd8e5ddf7903_TB2342_11 - y^2)dy

We can integrate this from y=0 to y=sqrt(11eaa948_cd04_6c30_84bc_dd8e5ddf7903_TB2342_11) to get the total volume.

∫[0, sqrt(11eaa948_cd04_6c30_84bc_dd8e5ddf7903_TB2342_11)] 2π(y-10)(11eaa948_cd04_6c30_84bc_dd8e5ddf7903_TB2342_11 - y^2)dy

Evaluating this integral will give us the answer, which is approximately 11eaa948_cd04_6c32_84bc_c32e43612fb4_TB2342_11. Therefore, the best choice is B.

Use Cylindrical Shells to Compute the Volume of the Solid