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Math 1314 Lesson 20 Evaluating Definite Integrals
We will sometimes need these properties when computing definite integrals.
Properties of Definite Integrals
Supposef andg are integrable functions. Then:
- (^) ∫aa f(x)dx= 0
- (^) ∫ (^) ab f(x)dx= −∫baf(x)dx
- (^) ∫ (^) ab cf(x)dx= c∫abf(x)dx
- (^) ∫ (^) ab [f(x)± g(x)]dx=∫abf(x)dx±∫bag(x)dx
- (^) ∫ ba f(x)dx=∫ (^) ac f(x)dx+∫cbf(x)dxwherea<c<b
We will need to use substitution to evaluate some problems:
Example 1: Evaluate ∫ ( − )
3
0
2 5 4 xx 3 dx
Example 2: Evaluate x e dx
1
0
∫^2 x^3
Example 3: Evaluate (^) ∫
2
1 3
2 dx 3 x 6
x
Example 6: The marginal daily profit function associated with production and sales of a video game is estimated to be
P' (x)= −. 0003 x^2 +. 04 x+ 17 wherex is the number of units
produced and sold daily and P' (x) is measured in dollars per
unit. Find the additional daily profit realizable if production and sales is increased from 200 units per day to 300 units per day.
The Average Value of a Function
We can use the definite integral to find the average value of a function.
Supposef is an integrable function on the interval [a, b]. Then
the average value off over the interval is (^) ∫ −
b b a a^ f(x)dx
. This
is what average value represents:
Example 7: Find the average value of f (x)= x over the
interval [1, 16].
Example 8: Find the average value of f (x)= x^2 − 3 x+ 5 on [2,
5].
