AB Calculus - Hardtke Assignment 6.2: Volumes of Revolution Name ______________________________

Due Date: Friday, 2/1

Think of the curve f(x) = √ over the domain [0, 4]. What happens if you revolve this curve about the x-axis? Example 1: How can we use calculus find the volume of this new solid?

Example 2: If we revolve the semi-circle y = √ about the x-axis, we obviously produce a _______________. Slicing this volume into more and more cross-sections gets our approximation closer to the actual volume.

*Now let’s use an integral to sum an infinite number of “disk” cross-sections.

Example 3: Find the volume of a solid if the region bounded by y = ex; x = 1 and x = 3 is revolved about the x-axis.

1. The region bounded by y = 2 – ½ x, x = 1 and x = 2 is revolved about the x-axis. Find the volume.

2. Find the volume of a solid formed if the region bounded by y = 1 – x2 and y = 0 is revolved about y = 0. Hint: how can you

use symmetry to make the limits on your integral easier to work with?

3. Find the volume of a solid created by revolving the region bounded by y = ¼ x2, x = 2 and y = 0 about the x-axis.

4. Careful: this region is revolved about the Y-axis! Find the volume if the region bounded by y = = x3, x = 0 and y = 8 is

revolved about the Y-axis.

AB Calculus - Hardtke Assignment 6.2: Volumes of Revolution Name ______________________________

Due Date: Friday, 2/1

Think of the curve f(x) = √ over the domain [0, 4]. What happens if you revolve this curve about the x-axis? Example 1: How can we use calculus find the volume of this new solid?

Example 2: If we revolve the semi-circle y = √ about the x-axis, we obviously produce a _______________. Slicing this volume into more and more cross-sections gets our approximation closer to the actual volume.

*Now let’s use an integral to sum an infinite number of “disk” cross-sections.

Example 3: Find the volume of a solid if the region bounded by y = ex; x = 1 and x = 3 is revolved about the x-axis.

5. The region bounded by y = 2 – ½ x, x = 1 and x = 2 is revolved about the x-axis. Find the volume.

6. Find the volume of a solid formed if the region bounded by y = 1 – x2 and y = 0 is revolved about y = 0. Hint: how can you

use symmetry to make the limits on your integral easier to work with?

7. Find the volume of a solid created by revolving the region bounded by y = ¼ x2, x = 2 and y = 0 about the x-axis.

8. Careful: this region is revolved about the Y-axis! Find the volume if the region bounded by y = = x3, x = 0 and y = 8 is revolved about the Y-axis.