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

308 STEP 4. Review the Knowledge You Need to Score High

You can use the symmetry of the region

and obtain the area= 2

∫ 2

0

(4−y^2 )dy.

An alternative method is to find the area

by setting up an integral with respect to

thex-axis and expressingx=y^2 asy=

√

x

andy=−

√

x.

8. (See Figure 13.8-7.)

A=

∫π

π/ 2

sin

(

x

2

)

dx

Letu=

x

2

anddu=

dx

2

or 2du=dx.

0 x

y

π π 2 π

2

x

2

π

x = 2 x = π

f(x) = sin( (

Figure 13.8-7

∫

sin

(

x

2

)

dx=

∫

sinu(2du)

= 2

∫

sinudu=−2 cosu+c

=−2 cos

(

x

2

)

+c

A=

∫π

π/ 2

sin

(

x

2

)

dx=

[

−2 cos

(

x

2

)]π

π/ 2

=− 2

[

cos

(

π

2

)

−cos

(

π/ 2

2

)]

=− 2

(

cos

(

π

2

)

−cos

(

π

4

))

=− 2

(

0 −

√

2

2

)

=

√

2

9. (See Figure 13.8-8.)
   Using the Disc Method:

V=π

∫ 3

0

(

x^2 + 4

) 2

dx

=π

∫ 3

0

(

x^4 + 8 x^2 + 16

)

dx

=π

[

x^5

5

+

8 x^3

3

+ 16 x

] 3

0

=π

[

35

5

+

8(3)^3

3

+16(3)

]

− 0 =

843

5

π

0 3

4

y y = x^2 + 4

Not to scale

x

Figure 13.8-8

10. 
    

    Area

    
    

∫k

1

1

x

dx=lnx]k 1 =lnk−ln 1=lnk.

Set lnk=1. Thus,elnk=e^1 ork=e.

11. (See Figure 13.8-9.)

1

0

• 1

y = 1

y = – 1

x

x = y^2 + 1

y

Figure 13.8-9