MAC 2233 - Worksheet 11

Name (PRINT): Meyline Michelin MAC 2233 Worksheet 11 f(x) = x^(a+1) / (a+1) = C Calculate the definite integral. Integral from 1 to 2 of (2x^3 - 3x + 5) dx (a) Find an antiderivative F(x) = 2x^4 / 4 - 3x^2 / 2 + 5x (b) Evaluate F(2) F(2) = 12 (c) Evaluate F(1) F(1) = 4 (d) Calculate the definite integral. Integral from 1 to 2 of (2x^3 - 3x + 5) dx = 8 Calculate the definite integral. Give exact answers. Integral from 0 to 3 of 3e^(-7x) dx (a) Find an antiderivative F(x) = -3/7 e^(-7x) (b) Evaluate F(0) F(0) = -3/7 (c) Evaluate F(-1) F(-1) = -3/7 e^7 (d) Calculate the definite integral. Integral from 0 to 3 of 3e^(-7x) dx = -3/7 (e^(-21) - 1) Compute Integral from -3 to -1 of (3/4x) dx 3/4 * Integral from -3 to -1 of (1/x) dx = 3/4 ln(x) f(-1) - f(-3) = 3/4 ln(-1) - 3/4 ln(-3) Integral (2x^3 - 3x + 5) dx Integral (2x^3 dx) = 2x^(3+1) / (3+1) = 2x^4 / 4 = x^4 / 2 Integral (3x dx) = 3x^(1+1) / (1+1) = 3x^2 / 2 Integral (5 dx) = 5 * x^1 / 1 = 5x F(2) = (2)^4 / 2 - 3(2)^2 / 2 + 5(2) = 16/2 - 12/2 + 10 = 8 - 6 + 10 = 12 F(1) = (1)^4 / 2 - 3(1)^2 / 2 + 5(1) = 1/2 - 3/2 + 5 = -1 + 5 = 4 F(2) - F(1) = 12 - 4 = 8 Integral from 0 to 3 of 3e^(-7x) dx e^(-7x) = 3e^(-7x) / -7 F(0) = -3/7 e^(-7(0)) = -3/7 F(-1) = -3/7 e^(-7(-1)) = -3/7 e^7 F(0) - F(-1) = -3/7 - (-3/7) e^7 = 3/7 (e^7 - 1) Using the graph of f compute the definite integrals (use known geometric formulas). width = 1 - (-2) = 3 height = 2 (a) Integral from -2 to 1 of f(x) dx = 3/2 = 3 (b) Integral from 1 to 2 of f(x) dx = (a+b)/2 * h = (1+2)/2 * 1 = 3/2 * 1 = 3/2 (c) Integral from 2 to 4 of f(x) dx = (a+b)/2 * h = (2+1)/2 * (-1) = 3/2 * (-1) = -3/2 (d) Integral from 0 to 6 of f(x) dx = 8/2 = 4 Using the graph of g compute the definite integrals. area = 3 area = 6 area = 1 (a) Integral from 1 to 3 of g(x) dx = 3 (b) Integral from 3 to 4 of g(x) dx = -1 (c) Integral from 1 to 8 of g(x) dx = 3 - 1 + 6 = 8 (d) (Circle one) Integral from 3 to 7 of g(x) dx = positive / negative / zero = 1 + 6 = 5*