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

∫

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 12.9-8.)
   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 12.9-8

10. 
    

    Area

    
    

∫k

1

1

x

dx=lnx]k 1 =lnk−ln 1=lnk.

Set lnk=1. Thuselnk=e^1 ork=e.

11. (See Figure 12.9-9.)

1

0

–1

y = 1

y = –1

x

x = y^2 + 1

y

Figure 12.9-9

Using the Disc Method:

V=π

∫ 1

− 1

(

y^2 + 1

) 2

dy

=π

∫ 1

− 1

(

y^4 + 2 y^2 + 1

)

dy

=π

[

y^5

5

+

2 y^3

3

+y

] 1

− 1

=π

[(

15

5

+

2(1)^3

3

+ 1

)

−

(

(−1)^5

5

+

2(−1)^3

3

+(−1)

)]

=π

(

28

15

+

28

15

)

=

56 π

15

Note: You can use the symmetry of the

region and find the volume by

2 π

∫ 1

0

(

y^2 + 1

) 2

dy.