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

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

Areas and Volumes 301

y

x

x = y

x = y^2

1

0

(1, 1)

Figure 13.4-14

Intersection points:y=x^2 ;x=y^2 ⇒y=±

√

x.

Setx^2 =

√

x⇒x^4 =x⇒x^4 −x= 0 ⇒x(x^3 −1)= 0 ⇒x=0orx= 1

x=0, y=0(0,0)

x=1, y=1(1,1).

Step 2: Determine the outer and inner radii of a washer.

The outer radius:

x=√yand inner radius:x=y^2.

Step 3: Set up an integral.

V=π

∫ 1

0

(

(√y)^2 −

(

y^2

) 2 )

dy

Step 4: Evaluate the integral.

Enter

∫ (

π∗

(

(√y)∧ 2 −(y∧ 2 )∧ 2

)

, y,0,1

)

and obtain

3 π

10

.

Thus, the volume of the solid is

3 π

10

.

13.5 Rapid Review

1. Iff(x)=

∫x

0

g(t)dtand the graph ofgis shown in Figure 13.5-1. Find f(3).

Answer: f( 3 )=

∫ 3

0

g(t)dt=

∫ 1

0

g(t)dt+

∫ 3

1

g(t)dt

= 0. 5 − 1. 5 =− 1