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

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

Def inite Integrals 253

The remaining properties are best illustrated in terms of the area under the curve of the

function, as discussed in the next section.

TIP • Do not forget that

∫− 3

0

f(x)dx=−

∫ 0

− 3

f(x)dx.

12.2 Fundamental Theorems of Calculus

Main Concepts:First Fundamental Theorem of Calculus, Second Fundamental

Theorem of Calculus

First Fundamental Theorem of Calculus

Iff is continuous on [a,b] andFis an antiderivative of fon [a,b], then

∫b

a

f(x)dx=F(b)−F(a).

Note:F(b)−F(a) is often denoted asF(x)

]b

a.

Example 1

Evaluate

∫ 2

0

(

4 x^3 +x− 1

)

dx.

∫ 2

0

(

4 x^3 +x− 1

)

dx=

4 x^4

4

+

x^2

2

−x

] 2

0

=x^4 +

x^2

2

−x

] 2

0

=

(

24 +

22

2

− 2

)

−(0)= 16

Example 2

Evaluate

∫π

−π

sinxdx.

∫π

−π

sinxdx=−cosx

]π

−π

=[−cosπ]−[−cos(−π)]

=[−(−1)]−[−(−1)]=(1)−(1)= 0

Example 3

If

∫k

− 2

(4x+1)dx=30,k>0, findk.

∫k

− 2

(4x+1)dx= 2 x^2 +x

]k

− 2 =

(

2 k^2 +k

)

−

(

2(−2)^2 − 2

)

= 2 k^2 +k− 6