5 Steps to a 5 AP Calculus BC 2019

Areas, Volumes, and Arc Lengths 283

Step 4. Evaluate the integral.

V=π

∫π/ 2

0

cosxdx=π[sinx]π/ 02 =π

(

sin

(

π

2

)

−sin 0

)

=π

Thus, the volume of the solid isπ.
Verify your result with a calculator.

Example 3

Find the volume of the solid generated by revolving about they-axis the region in the first
quadrant bounded by the graph ofy=x^2 , they-axis, and the liney=6.

Step 1. Draw a sketch. (See Figure 12.4-9.)

6

0

y

y = 6

x = √y

x

Figure 12.4-9

Step 2. Determine the radius from a cross section.

y=x^2 ⇒x=±

√

y

x=

√

yis the part of the curve involved in the region.

r=x=

√

y

Step 3. Set up an integral.

V=π

∫ 6

0

x^2 dy=π

∫ 6

0

(

√

y)^2 dy=π

∫ 6

0

ydy

Step 4. Evaluate the integral.

V=π

∫ 6

0

ydy=π

[

y^2

2

] 6

0

= 18 π

The volume of the solid is 18π.
Verify your result with a calculator.