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5.4 Indefinite Integrals and the Net Change Theorem 1. Definition (Indefinite integral) If f (x) is a continuous function, the general form of the antiderivatives of f is called the indefinite integral of f and it is denoted by Z f (x)dx Notice the lack of lower and upper bounds compared to a definite integral. 2. Formulas We redo the formulas for antiderivatives in terms of indefinite integrals : • (Linear property of an integral) For any constants α, β, and continuous functions f (x) and g(x), Z Z Z αf (x) + βg(x)dx = α f (x)dx + β g(x)dx Z • 1dx = x + C Z • xn dx = xn+1 + C (n 6= −1) n+1 Z sin xdx Z cos xdx • Z sec2 xdx (Summary) Integrals Z sec x tan xdx Z csc x cot xdx • Z csc2 xdx = − cos x + C = sin x + C = tan x + C Indefinite integral : Definite integral : = sec x + C = − csc x + C = − cot x + C Z f (x)dx Z : anti-derivatives of f (x) : the area under the graph y = f (x) on [a, b]. b f (x)dx a Fundamental Theorem Indefinite integral Definite integral of Calculus Z Z a 3. Example Find the indefinite integral : Z cos θ dθ sin2 θ Z 1 + 3 − csc2 xdx x2 (a) (b) Answers : (a) − csc θ + C, (b)− b f (x)dx = F (b) − F (a) f (x)dx = F (x) + C 1 + 3x + cot x + C x 1 4. Example Evaluate the indefinite integral and interpret the geometric meaning : Z 4 (a) √ t(1 + t)dt 1 Z 4 r (b) 1 Z π/3 (c) −π/6 Z 5 dx x sin θ + sin θ tan2 θ dθ sec2 θ 3π/2 | sin x|dx (d) 0 Answers : (a) 256 15 √ (b) 2 5 (c) −1 + 2 √ 3 (d) 3 5. Theorem (The Net Change Theorem) When a changing quantity Q(t) changes by a rate r(t), the net change of the quantity during a ≤ t ≤ b is Z Q(b) − Q(a) = b r(t)dt a 100 Z 6. Example (a) If a particle moves on x−axis with the velocity v(t)unit/sec, v(t)dt represents displacement of 0 the particle after 100 seconds. 100 Z |v(t)|dt represents the total travel (b) If a particle moves on x−axis with the velocity v(t)unit/sec, 0 distance during the first 100 seconds. Z (c) If water leaks from a water tank by a rate r(t) gal/min, 60 r(t)dt represents the total water loss 0 during the first 60minutes. Z 20 (d) If ρ(x)(gram/cm) is the density of a rod at the point x cm from the left end, the mass of a portion of the rod between 10cm and 20cm from left end. ρ(x)dx represents 10 (e) If b(t)(persons/month) is the number of births per month(called birth rate) on earth and d(t)(persons/month) is the number of deaths per month(called death rate) on earth, Z 12 b(t) − d(t)dt represents the total change of world population during the first 12 months since 0 the observation. 2 7. Example (a) The acceleration of a particle moving along x−axis is a(t) = 2t + 3 (unit/sec2 ) When the initial velocity v(0) = −4, find the total travel distance during 0 ≤ t ≤ 2. (b) Oil leaks from the bottom of a container at a rate of √ r(t) = 10 − t (liters/min) Find the total amount of oil loss during the first 25 minutes. 2 Z Answers : 2 Z 1 0 0 2 Z 2 −(t + 3t − 4)dt + t + 3t − 4 dt = (a) 1 Figure for (a) 3 2 t + 3t − 4dt = 5 25 Z 10 − (b) 0 √ tdt = 500 liters 3