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AP Calculus AB: 10.6.1 Solids of Revolution

Mathematics7 CardsCreated 3 months ago

This flashcard set introduces solids of revolution formed by rotating a plane region about a line. It explains the disk method for calculating volume by integrating the area of circular cross-sections, emphasizing visualization, setting up integrals with the radius function, and solving examples involving common functions.

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Solids of Revolution

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

Solids of Revolution

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

  • Some solids can be described by moving regions through space, as well as by slicing the solid into pieces.

  • Consider the plan...

What is the volume of the solid that is generated by revolving the plane region bounded by y = x 2, y = 0, and x = 1 about the x‑axis?

π/5

What is the volume of the solid that is generated by revolving the plane region bounded by y=cos x and y=0 about the x-axis from x=0 to x=π2?

π^2/4

Which of the following is the definition of a solid of revolution?

A solid of revolution is the object formed by revolving a region of a plane around a line.

Which of these integrals defines the volume of the solid that is generated by revolving the plane region bounded by y = f (x) and y = 0 about the x‑axis from x = a to x = b ?

∫^b _a π[f(x)]^2dx

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AP Calculus AB: 10.6.1 Solids of Revolution
TermDefinition

Solids of Revolution

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

  • Some solids can be described by moving regions through space, as well as by slicing the solid into pieces.

  • Consider the plane region given to the left.

  • What happens if you rotate that region around the x-axis?

  • To visualize the solid, think of the region as though it were connected to the x-axis on a hinge. As the region moves through space around the hinge, the space it passes through makes up the solid.

  • A solid defined in this way is called a solid of revolution.

  • To find the volume of a solid of revolution you can sometimes divide the region into slices. Each slice resembles a disk, so this method is called the disk method.

  • To find the volume, just integrate the areas of the disks across the given interval.

  • The radius of a given disk is equal to the height of the original region. The area of a disk equals the area of a circle.

  • Once you find the area, just integrate. Notice that sometimes you can find shortcuts in the integral based upon symmetry or other properties of the region.

  • Setting up the integral is the tough part of finding volumes. Once you have the integral, evaluating it is a piece of cake.

What is the volume of the solid that is generated by revolving the plane region bounded by y = x 2, y = 0, and x = 1 about the x‑axis?

π/5

What is the volume of the solid that is generated by revolving the plane region bounded by y=cos x and y=0 about the x-axis from x=0 to x=π2?

π^2/4

Which of the following is the definition of a solid of revolution?

A solid of revolution is the object formed by revolving a region of a plane around a line.

Which of these integrals defines the volume of the solid that is generated by revolving the plane region bounded by y = f (x) and y = 0 about the x‑axis from x = a to x = b ?

∫^b _a π[f(x)]^2dx

What is the volume of the solid that is generated by revolving the plane region bounded by y=sin2x and y=0 about the x-axis from x=0 to x=π/2?

π^2/4