10.6.3 A Transcendental Example of the Disk Method

Question 1

A Transcendental Example of the Disk Method

Answer 1

• Revolving a plane region about a line forms a solid of revolution.
• Using the disk method, the volume V of a solid of revolution is given by , where R(x) is the radius of
the solid of revolution with respect to x.

Question 2

note

Answer 2

• Find the volume of the solid of revolution described on the left.
• Start by sketching the region in question. Notice that the sketch of the square root of sine looks like a distorted version of the sine curve.
• Once you have the region, you have to visualize what the solid of revolution will look like. Since you are revolving around the x-axis, the solid will look sort of like a football.
• Slicing the solid horizontally doesn’t look like a good idea. But vertical slices would give circular disks. Therefore, you can find the volume of the solid by using the disk method.
• The thickness of each disk is a small value in the x-direction. Call it dx. The disks run from 0 to π, which are the limits of integration.
• The radius of each disk is equal to the y-value of the function. Square the radius and multiply by π.
• Simplify the product so it is easier to integrate.
• Notice that the hardest step in these volume problems is setting up the integral. The actual calculus is not that tough. It’s the set-up that is a little confusing.

Question 3

Consider the solid of revolution generated by rotating the area bounded by y=√cos x, the x-axis,x=0 and x=π/2 around the x-axis.What will be the radius of the disk in the formula to determine the volume using disks?

Answer 3

√cos x

Question 4

What is the volume of the solid of revolution generated by rotating the area bounded by y=√sinx, the x-axis, x=π/4, and x=3π/4 around the x-axis?

Answer 4

π√2 units^3

Question 5

Consider the solid of revolution generated by rotating the area bounded by y=√cosx, the x-axis,x=0 and x=π/2 around the x-axis.What will be the lower limit of integration in the formula to determine the volume using disks?

Answer 5

0

Question 6

What is the volume of the solid of revolution generated by rotating the area bounded by y=√cosx, the x-axis, x=0, and x=π/2 around the x-axis?

Answer 6

π units^3

Question 7

Consider the solid of revolution generated by rotating the area bounded by y=√cosx, the x-axis,x=0 and x=π/2 around the x-axis.What will be the variable of integration in the formula to determine the volume using disks?

Answer 7

x

Question 8

Consider the solid of revolution generated by rotating the area bounded by y=√cosx, the x-axis,x=0 and x=π/2 around the x-axis.What will be the upper limit of integration in the formula to determine the volume using disks?

Answer 8

π/2