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

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

Areas and Volumes 291

Example 2

Find the volume of a pyramid whose base is a square with a side of 6 feet long, and a height
of 10 feet. (See Figure 13.4-3.)

x

y

0

10

6

6

s

x

3

10

Figure 13.4-3

Step 1: Find the area of a cross sectionA(x). Note each cross section is a square of side 2s.

Similar triangles:

x

s

=

10

3

⇒s=

3 x

10

.

A(x)=(2s)^2 = 4 s^2 = 4

(

3 x

10

) 2

=

9 x^2

25

Step 2: Set up an integral.

V=

∫ 10

0

9 x^2

25

dx

Step 3: Evaluate the integral.

V=

∫ 10

0

9 x^2

25

dx=

3 x^3

25

] 10

0

=

3 ( 10 )^3

25

− 0 = 120

The volume of the pyramid is 120 ft^3.