Ap Calculus Notes

AP CALCULUS NOTES SECTION 7.2 VOLUMES BY SLICING; DISKS AND WASHERS

In this section, we define volumes of solids whose cross sections are plane regions.

Definition: A cross section of a solid S is the plane region formed by intersecting S with a plane.

For example, the volume V of a right circular cylinder of radius r can be expressed as V A  h  area of a cross section  height .

n n * * This equation as well as the lim f xk  x ( or lim f xk x k ), which is used to n    maxx  0    k 1 k k 1 determine the area under a curve, forms the basis for defining the volumes of many solids. For solids that are not cylindrical, we use the method of slicing.

A.) The volume of a solid can be obtained by integrating the cross-sectional area from one end of a solid to the other. Volume Formula #1: Let S be a solid bounded by 2 parallel planes perpendicular to the x-axis at x a and x b . If for each x in a, b , the cross-sectional area of S perpendicular to the x-axis is A x , where A is a continuous and integrable function, then the volume of the solid is n * b Vlim A xk  x k  A x dx . n     a k 1 Note: Remember that A x represents the area of a moving cross section obtained by slicing through x perpendicular to the x-axis.

Volume Formula #2: Let S be a solid bounded by 2 parallel planes perpendicular to the y-axis at y c and y d . If for each y in c, d , the cross-sectional area of S perpendicular to the y-axis is A y , where A is a continuous and integrable function, then the volume of the solid is d V A y dy .  c Ex.1.) A pyramid 3m high has a square base that is 3m on a side. The cross section of the pyramid perpendicular to the altitude x m down from the vertex is a square x m on a side as shown in the figure below. Find the volume of the pyramid. Steps: 1.) Sketch the solid and a typical cross section. 2.) Find a formula for A x .

3.) Find the limits of integration.

4.) Integrate A x using the FTC. B.) Solid of Revolution – a solid generated by revolving a plane region about an axis or line of revolution that lies in its plane.

1.) Disk Method for a solid revolved about the x -axis on the interval a, b . The cross section 2 formed creates a circular disk of radius y R x  f x where the A   radius . b b 2 V R x 2 dx    f x  dx a   a   Ex.2.) Find the volume of the solid that is obtained when the region under the curve y x over the interval 0,4 is revolved about the x-axis.

Ex.3.) Find the volume of the sphere that is generated by rotating the circle x2 y 2  a 2 about the x-axis.

Not all solids of revolution have solid interiors. Some solids have holes; this implies the cross sections are “washer-shaped” instead of “disk-shaped”. 2.) Washer Method for a solid revolved about the x -axis on the interval a, b . 2 2 Outer Radius: R x  f x A outer radius    inner radius b2 2 b 2 2 Inner Radius: r x  g x V R x    r x  dx    f x    g x  dx a     a       Ex.4.) Find the volume of the solid generated when the region enclosed by the graphs of y x2 1 and y  x  3 is revolved about the x-axis.

3.) Disk Method for a solid revolved about the y -axis on the interval c, d . d d 2 V R y 2 dy    u y  dy c   c   2 Ex.5.) Find the volume of the solid generated by revolving the region between x  and x  0 y over the interval 1,4 about the y-axis.

4.) Washer Method for a solid revolved about the y -axis on the interval c, d . d2 2 d 2 2 V R y    r y  dy    w y    v y  dy c     c       Ex.6.) Find the volume of the solid generated by revolving the region between y x2 and y 2 x about the y-axis. 5.) Solid generated by revolving about a horizontal or vertical line. Ex.7.) Find the volume of the solid generated by revolving the region bounded by y x , y 1 , and x  4 about the line y 1.

0,5 0,3

Ex.8.) Find the volume of the solid generated by revolving the region between the parabola x y2 1 and the line x  3 about the line x  3.

0,4  4,4 Ex.9.) Find the volume of the solid generated by revolving the region between y x and y x2 a.) about the line y  2 .

.5,1.5   .5,2.5

b.) about the line x  1.

1.5,1.5   .5,1.5