MATH 151 Engineering Mathematics I Fall 2017, WEEK 14 JoungDong Kim Week 14 Section 5.4, 5.5, 6.1, Indefinite Integrals and the Net Change Theorem, The Substitution Rule, Areas Between Curves. Section 5.4 Indefinite Integrals and the Net Change Theorem The notation f(x) dx is traditionally used for an antiderivative of f and is called an indefinite integral. Thus f(x) dx = F(x) means F (x) = f(x) Note. A definite integral function. b a f(x) dx is a number, whereas an indefinite integral f(x) dx is a 1
Table of Indefinite Integrals 1. 2. 3. cf(x) dx = c f(x) dx [f(x)+g(x)] dx = k dx = kx+c f(x) dx+ 4. x n dx = xn+1 +C (n 1) n+1 1 5. dx = ln x +C x 6. e x dx = e x +C 7. 8. 9. 10. 11. 12. 13. 14. 15. b x dx = bx lnb +C sinx dx = cosx+c cosx dx = sinx+c sec 2 x dx = tanx+c csc 2 x dx = cotx+c secxtanx dx = secx+c cscxcotx dx = cscx+c 1 x 2 +1 dx = tan 1 x+c 1 1 x 2 dx = sin 1 x+c g(x) dx 2
Ex.1) Find the general indefinite integral ( a) x 6 2x 5 x 3 + 2 ) dx 7 b) (10x 4 2sec 2 x) dx c) cosθ sin 2 θ dθ d) 4 x5 dx e) 1+ x+x x dx f) ( 2x 3 6x+ 3 ) x 2 +1 dx 3
Ex.2) Evaluate the integral. 2 ( a) 2x 3 6x+ 3 ) x 2 +1 0 dx b) π 0 (5e x +3sinx) dx c) 2 0 (2x 3)(4x 2 +1) dx 4
Net Change Theorem The integral of a rate of change is the net change: b a F (x) dx = F(b) F(a) Ex.3) A particle moves along a line so that its velocity at time t is v(t) = t 2 t 6 (measured in meters per second). a) Find the displacement of the particle during the time period 1 t 4. b) Find the distance traveled during this time period. 5
Ex.4) Water flows from the bottom of a storage tank at a rate of r(t) = 200 4t liters per minutes, where 0 t 50. Find the amount of water that flows the tank during the first 10 minutes. 6
Section 5.5 The Substitution Rule The Substitution Rule If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du Ex.5) Evaluate the integral. a) x 3 cos(x 4 +2) dx b) 2x+1 dx 7
c) x 1 4x 2 dx d) e 5x sin(e 5x ) dx e) x 5 1+x 2 dx f) tanx dx 8
The Substitution Rule for Definite Integrals If g is continuous on [a,b] and f is continuous on the range of u = g(x), then b f(g(x))g (x) dx = g(b) a g(a) f(u) du Ex.6) Evalute the definite integral. a) 4 0 2x+1 dx b) 2 1 dx (3 5x) 2 c) e 1 lnx x dx 9
Integrals of Symmetric Functions Suppose f is continuous on [ a,a]. a) If f is even [f( x) = f(x)], then b) If f is odd [f( x) = f(x)], then a a a a a f(x) dx = 2 f(x) dx = 0. 0 f(x) dx. Ex.7) Evaluate the definite integral. a) 2 2 (x 6 +1) dx b) 1 1 (x 3 x) dx Ex.8) If f is continuous and 9 0 f(x) dx = 4, find 3 0 xf(x 2 ) dx. 10
Section 6.1 Area Between Curves The area A of the region bounded by the curves y = f(x), y = g(x), and the lines x = a, x = b, where f and g are continuous and f(x) g(x) for all x in [a,b], is A = b a [f(x) g(x)] dx Ex.9) Find the area of the region bounded above by y = e x, bounded below by y = x, and bounded on the sides by x = 0 and x = 1. Ex.10) Find the area of the region enclosed by the parabolas y = x 2 and y = 2x x 2. 11
The area between the curves y = f(x) and y = g(x) and between x = a and x = b is A = b a f(x) g(x) dx Ex.11) Find the area of the region bounded by the curves y = sinx, y = cosx, x = 0, and x = π 2. 12
Ex.12) Find the area of the region. a) y = 1 x, y = 1 x 2, x = 2. b) y = cosx, y = 2 cosx, 0 x 2π. 13
Ex.13) Find the area enclosed by the line y = x 1 and the parabola y 2 = 2x+6. 14
Ex.14) The graphs of two functions are shown with the areas of the regions between the curves indicated. a) What is the total area between the curves for 0 x 5? b) What is the value of 5 [f(x) g(x)] dx? 0 Ex.15) Find the area of the region bounded by the parabola y = x 2, the tangent line to this parabola at (1,1), and the x-axis. 15
Ex.16) Find the area of the region bounded by the three lines below. y = 1 x, y = 7x, y = 3x+10 3 Ex.17) Find the area of the region enclosed by the curves y = x 3 2x, y = 7x. 16