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 need to integrate with respect to y, since the region is being revolved around a horizontal line.

First, we need to find the limits of integration. The region is bounded by the curves y = x^2, y = 0, and x = 1. When y = 10, we have:

10 = x^2
x = ±√10

Since the region is bounded by x = 1, we only need to integrate from y = 0 to y = 10.

Now, we need to find the radius and height of each cylinder. For this particular problem, it is easiest to use the axis of revolution (y = 10) as the height. Then, the radius will be the distance from the axis of revolution to the curve y = x^2. Since we are revolving around y = 10, this distance is:

r = 10 - x^2

So, the volume of each cylinder is:

dV = 2πrh dy
dV = 2π(10 - x^2)(x) dy
dV = 20πx dy - 2πx^3 dy

Integrating from y = 0 to y = 10, we have:

V = ∫(20πx - 2πx^3) dy from y = 0 to y = 10
V = 20π∫x dy - 2π∫x^3 dy from y = 0 to y = 10
V = 20π[x^2/2] - 2π[x^4/4] from y = 0 to y = 10
V = 10π(100 - 5√10)

Simplifying, we get:

V = 50π(2√10 - 1)
V ≈ 409.14

Therefore, the answer is B) 11eaa88b_918e_64db_a696_1d4fb74be5ee_TB5869_11.

Use Cylindrical Shells to Compute the Volume of the Solid