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

MA 3972-MA-Book April 11, 2018 15:57

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

Example 3

Evaluate

∫− 1

− 8

(

√ (^3) y+ 1
√ (^3) y
)
dy.
Rewrite:
∫− 1
− 8
(
y^1 /^3 +

1

y^1 /^3

)

dy=

∫− 1

− 8

(

y^1 /^3 +y−^1 /^3

)

dy

=

y^4 /^3

4 / 3

+

y^2 /^3

2 / 3

]− 1

− 8

=

3 y^4 /^3

4

+

3 y^2 /^3

2

]− 1

− 8

=

(

3(−1)^4 /^3

4

+

3(−1)^2 /^3

2

)

−

(

3(−8)^4 /^3

4

+

3(−8)^2 /^3

2

)

=

(

3

4

+

3

2

)

−(12+6)=

− 63

4

.

Verify your result with a calculator.

TIP • You may bring up to 2 (but no more than 2) approved graphing calculators to the

exam.

Definite Integrals Involving Absolute Value

Example 1

Evaluate

∫ 4

1

∣∣

3 x− 6

∣∣

dx.

Set 3x− 6 =0;x=2; thus,

∣∣

3 x− 6

∣∣

=

{

3 x−6ifx≥ 2

−(3x−6) ifx< 2

.

Rewrite:

∫ 4

1

∣∣

3 x− 6

∣∣

dx=

∫ 2

1

−(3x−6)dx+

∫ 4

2

(3x−6)dx

=

[

− 3 x^2

2

+ 6 x

] 2

1

+

[

3 x^2

2

− 6 x

] 4

2

=

(

−3(2)^2

2

−6(2)

)

−

(

−3(1)^2

2

−6(1)

)

+

(

3(4)^2

2

−6(4)

)

−

(

3(2)^2

2

−6(2)

)

=(− 6 +12)−

(

−

3

2

+ 6

)

+(24−24)−(6−12)

= 6 − 4

1

2

+ 0 + 6 =

15

2

.

Verify your result with a calculator.