4.5. Evaluating Definite Integrals http://www.ck12.org
Using the limit definition we found that∫ 03 x^3 dx=^814 .We now can verify this using the theorem as follows:
We first note thatx^4 /4 is an antiderivative off(x) =x^3 .Hence we have
∫ 3
0 x(^3) dx=x^4
4
] 3
0=^814 −^04 =^814.
We conclude the lesson by stating the rules for definite integrals, most of which parallel the rules we stated for the
general indefinite integrals.
∫a
∫abf(x)dx=^0
a f(x)dx=−∫a∫b b f(x)dx
a k·f(x)dx=k∫b∫ a f(x)dx
b
a[f(x)±g(x)]dx=∫b
a f(x)dx±∫b
∫b a g(x)dx
a f(x)dx=∫c
a f(x)dx+∫b
c f(x)dxwherea<c<b.Given these rules together with Theorem 4.1, we will be able to solve a great variety of definite integrals.
Example 2:
Compute∫−^22 (x−√x)dx.
Solution:
∫ 4
1 (x−√x)dx=∫^4
1 xdx−∫ 4
1√xdx=x^2
2] 4
1 −
2
3 x32 ]^4
1 =
(
8 −^12
)
−^23 ( 8 − 1 ) =^152 −^143 =^176.
Example 3:
Compute∫
π 2
0 (x+cosx)dx.
Solution:
∫π 2
0 (x+cosx)dx=∫π 2
0 (x)dx=∫ π 2
0 (cosx)dx=x^2
2∣∣
∣
π 2
0 +sinx
1∣∣
∣
π 2
0 =π^2
8 +^1 =π^2 + 8
8.Lesson Summary
- We used antiderivatives to evaluate definite integrals.
- We used the Mean Value Theorem for integrals to solve problems.
- We used general rules of integrals to solve problems.
Proof of Theorem 4.1