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MATH 131-503 Fall 2015 c Wen Liu 5.3 5.3 Evaluating Definite Integrals Evaluation Theorem: If f is continuous on the interval [a, b], then Z b f (x)dx = F (b) − F (a) a where F is any antiderivative of f , that is, F 0 = f . Table of Indefinite Integrals Z • Z cf (x)dx = c Z • Z • Z f (x)dx xn+1 x dx = + C, n 6= −1 n+1 n x x e dx = e + C sin xdx = − cos x + C Z • 2 sec xdx = tan x + C sec x tan xdx = sec x + C Z • Z • Z • f (x)dx ± g(x)dx 1 dx = ln |x| + C x ax dx = ax +C ln a • cos xdx = sin x + C Z • csc2 xdx = − cot x + C Z Z • (f (x) ± g(x)) dx = Z Z Z • • Z 1 dx = tan−1 x + C 2 x +1 • csc x cot xdx = − csc x + C Z • 1 √ dx = sin−1 x + C 2 1−x Examples: Compute each of the following. Z 1 3 1. 2x3 − 6x − 2 + dx x x Page 1 of 3 MATH 131-503 Fall 2015 Z 5.3 c Wen Liu 3 (x2 − 3)(x + 1)dx 2. 1 Z 2 |2x − 1|dx 3. 0 Z 4. (p. 363) 1 2 (x − 1)3 dx x2 Page 2 of 3 MATH 131-503 Fall 2015 Z 5. 5.3 c Wen Liu (6ex + ax − 2 sec2 x)dx Net Change Theorem: The integral of a rate of change is the net change Z b F 0 (x)dx = F (b) − F (a) a This principle can be applied to all of the rates of change in the natural and social sciences that we discussed in Section 3.8. Here are a few instances of this idea: • If V (t) is the volume of water in a reservoir at time t, then its derivative V 0 (t) is the rate at which water flows into the reservoir at time t. So Z t2 V 0 (t)dt = V (t2 ) − V (t1 ) t1 is the change in the amount of water in the reservoir between time t1 and time t2 . • If we want to calculate the distance the object travels during the time interval, we have to consider the intervals when v(t) ≥ 0 (the particle moves to the right) and also the intervals when v(t) ≤ 0 (the particle moves to the left). In both cases the distance is computed by integrating |v(t)|, the speed. Therefore Z t2 |v(t)|dt = total distance travled t1 Example: Water flows from the bottom of a storage tank at a rate of r(t) = 300 − 6t liters per minute, where 0 ≤ t ≤ 50. Find the amount of water that flows from the tank during the first 30 minutes. Page 3 of 3