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AP Calculus AB: 10.6.3 A Transcendental Example of the Disk Method

Mathematics8 CardsCreated 3 months ago

This set explores finding volumes of solids of revolution involving transcendental functions like sine and cosine using the disk method. It emphasizes careful sketching, visualizing the solid, setting up integrals with the correct radius function, and integrating over specified limits to compute volume.

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A Transcendental Example of the Disk Method

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.

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Key Terms

Term
Definition

A Transcendental Example of the Disk Method

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 , ...

note

  • Find the volume of the solid of revolution described on the left.

  • Start by sketching the region in question. Notice that the...

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?

√cos x

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?

π√2 units^3

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?

0

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?

π units^3

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AP Calculus AB: 10.6.3 A Transcendental Example of the Disk Method
TermDefinition

A Transcendental Example of the Disk Method

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.

note

  • 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.

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?

√cos x

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?

π√2 units^3

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?

0

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?

π units^3

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?

x

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?

π/2