Review - Volumes of Revolution

Front

Define Volume of Revolution.

Back

The volume of revolution is the volume of a solid formed by rotating a two-dimensional shape around an axis.

Front

What is the formula for the volume of revolution around the x-axis using the disk method?

Back

V = π ∫[a to b] (f(x))^2 dx, where f(x) is the function being rotated.

Front

What is the formula for the volume of revolution around the y-axis using the shell method?

Back

V = 2π ∫[a to b] x f(x) dx, where f(x) is the function being rotated.

Front

Explain the disk method for finding volumes of revolution.

Back

The disk method involves slicing the solid into thin disks perpendicular to the axis of rotation and summing their volumes.

Front

Explain the shell method for finding volumes of revolution.

Back

The shell method involves slicing the solid into cylindrical shells parallel to the axis of rotation and summing their volumes.

Front

Find the volume of the solid generated by revolving the area bounded by y = x^2 and the x-axis from [0, 2] around the x-axis.

Back

32π/5

Front

Find the volume of the solid generated by revolving the area bounded by y = x^2 and the x-axis from [0, 2] around the y-axis.

Back

8π

Front

Find the volume of the solid of revolution obtained by rotating the region bounded by y = x^3 + 1, x = 1 and y = 1 about the y-axis.

Back

2π/5

Front

What is the volume of the solid formed in the second quadrant bounded by the curves y = 9 – x^2, the x-axis and the y-axis when it is rotated about the line x = 1?

Back

153π/2

Front

If the region enclosed by y = 3x^2, y = 3 is revolved about the x-axis, what is the volume of the solid generated?

Back

72π/5