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

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

Areas and Volumes 295

Step 2: Determine the radius of a disc from a cross section.

r= f(x)=

√

x− 1

Step 3: Set up an integral.

V=π

∫ 5

1

(f(x))^2 dx=π

∫ 5

1

(√

x− 1

) 2

dx

Step 4: Evaluate the integral.

V=π

∫ 5

1

(√

x− 1

) 2

dx=π[(x− 1 )]^51 =π

[

x^2

2

−x

] 5

1

=π

((

52

2

− 5

)

−

(

12

2

− 1

))

= 8 π

Verify your result with a calculator.

Example 2

Find the volume of the solid generated by revolving about thex-axis the region bounded by
the graph ofy=

√

cosxwhere 0≤x≤

π

2

, thex-axis, and they-axis.

Step 1: Draw a sketch. (See Figure 13.4-8.)

y

x

y = cos x

0

1

π

2

Figure 13.4-8

Step 2: Determine the radius from a cross section.

r= f(x)=

√

cosx

Step 3: Set up an integral.

V=π

∫π/ 2

0

(√

cosx

) 2

dx=π

∫π/ 2

0

cosxdx