5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 16:1

Areas and Volumes 297

Example 4

Using a calculator, find the volume of the solid generated by revolving about the liney=8,
the region bounded by the graph ofy=x^2 +4, and the liney=8.

Step 1: Draw a sketch. (See Figure 13.4-10.)

• 2 2

4

8

0

y = 8

y = x^2 + 4

y

x

Figure 13.4-10

Step 2: Determine the radius from a cross section.

r= 8 −y= 8 −(x^2 +4)= 4 −x^2

Step 3: Set up an integral.
To find the intersection points, set 8=x^2 + 4 ⇒x=±2.

V=π

∫ 2

− 2

(

4 −x^2

) 2

dx

Step 4: Evaluate the integral.

Enter

∫ (

π

(

4 −x^2

) 2

, x, −2, 2

)

and obtain

512

15

π.

Thus, the volume of the solid is

512

15

π.

Verify your result with a calculator.

Example 5

Using a calculator, find the volume of the solid generated by revolving about the liney=−3,
the region bounded by the graph ofy=ex, they-axis, and the linesx=ln 2 andy=−3.

Step 1: Draw a sketch. (See Figure 13.4-11.)