Calculus Question

Calculus: Learning Worksheets Chapter 5 Name ________________________________ Date ______________ Class ____________ Section 5-4 The Definite Integral Goal: To calculate the values of definite integrals using the properties. Theorem: Limits of Left and Right Sums If f ( x)  0 and is either increasing or decreasing on [a, b], then its left and right sums approach the same real number as n  . Theorem: Limit of Riemann Sums If f is a continuous function on [a, b], then the Riemann sums for f on [a, b] approach a real number limit I as n  . Definition: Definite Integral Let f be a continuous function on [a, b]. The limit I of Riemann sums for f on [a, b] is called the definite integral of f from a to b and is denoted as ab f ( x)dx. Properties: Definite Integrals 1. aa f ( x)dx  0 2. ab f ( x)dx   ba f ( x)dx 3. ab kf ( x)dx  k ab f ( x)dx, k a constant 4. ab [ f ( x)  g ( x)]dx  ab f ( x)dx  ab g ( x)dx 5. ab f ( x)dx  ac f ( x)dx  cb f ( x)dx 181 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 In Problems 1 and 2, calculate the indicated Riemann sum Sn for the function f ( x) = 7 + x 2 . 1. Partition [–1, 9] into five subintervals of equal length, and for each subinterval [ xk 1, xk ], let ck  ( xk 1  xk ) / 2. 2. Partition [–4, 8] into four subintervals of equal length, and for each subinterval [ xk 1, xk ], let ck  (2 xk 1  xk ) / 3. 182 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 In Problems 3–7, calculate the definite integral, given that 6 ∫0 x dx = 18 6 2 ∫0 x dx = 72 9 2 ∫6 x dx = 171 3. 04 3x dx 4. 04 (2 x  x 2 ) dx 183 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 5. 07 2x 2 dx 6. 60 7x dx 7. 77 ( x 2  2 x  15) dx 184 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 Name ________________________________ Date ______________ Class ____________ Section 5-5 The Fundamental Theorem of Calculus Goal: To use the fundamental theorem of calculus to solve problems. Theorem 1: Fundamental Theorem of Calculus If f is a continuous function on [a, b], and F is any antiderivative of f, then b a f ( x)dx  F (b)  F (a) Definition: Average Value of a Continuous Function f over [a, b] 1 b  f ( x)dx ba a In Problems 1–7, evaluate the integrals. 1. ∫26 (8 x3 + 3) dx 185 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 2. ∫05 (2e x ) dx 3. ∫77 (2 x3 + 5 x 2 − x + 27) dx 22 4. 13 5 dx x3 186 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 5. 01 2 x3 (3x 4  2)6 dx 6. 12 3x 2  2 x  5 x3  x 2  5 x  3 dx 187 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 7. 12 3 x 6 x 2  1 dx In Problems 8 and 9, find the average value of the function over the given interval. 8. f ( x)  8 x3  16 x  4; [1, 6] 188 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 9. g ( x)  2e 0.3 x Chapter 5 [0,10] 10. The total cost (in dollars) of manufacturing x units of a product is C ( x) = 50, 000 + 400 x. a) Find the average cost per unit if 400 units are produced. [Hint: Recall that C ( x) is the average cost per unit.] b) Find the average value of the cost over the interval [0, 400]. 189 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 5 11. A company manufactures a product and the research department produced the marginal cost function C ‘( x)  300  x 4 0  x  800, where C ‘( x) is in dollars and x is the number of units produced per month. Compute the increase in cost going from a production level of 400 units per month to 800 units per month. Set up a definite integral and evaluate it. 190 Copyright © 2019 Pearson Education, Inc.