Flashcards in 10.6.1 Solids of Revolution Deck (7):
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.
- 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?
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?
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