Calculus 4-R Unit 4 Integration Review Problems. Evaluate 6 1.

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Presentation transcript:

Calculus 4-R Unit 4 Integration Review Problems

Evaluate 6 1

Review Problems -2 1 Evaluate 2

Review Problems Find the average value of f(x) = 3x on the interval [0, 2]. 2 Find the average value of y = x 3 over the interval [0, 2]. 2 3

Review Problems Find the area of the region bounded by y = (x - 1) 2 + 1, the x-axis, x = -1, and x = 2. 6 Find the value of c guaranteed by the Mean Value Theorem for Integrals for on the interval [1, 4]. 2 4

Review Problems Consider Find and 5

Review Problems Evaluate 6

Review Problems Evaluate 7

Review Problems Evaluate 8

Review Problems Evaluate 9

Review Problems Evaluate 10

Review Problems Find an expression in a and b for the value of the definite integral 11

Review Problems Consider the integral, Determine new upper and lower limits of integration using the substitution u = 4x Upper, 97; lower, 5 Use the Trapezoidal Rule, with n = 4, to approximate

Review Problems Use the Trapezoidal Rule, with n = 4, to approximate the area of the region bounded by the graphs of y = sin x and y = 0 on the interval [0, π ]. 13

Review Problems Find the function, y = f(x), if and f(1) = 3 y = x 2 - x + 3 Find y = f(x) if and f(0) = 2. 14

Review Problems Evaluate 15

Review Problems Evaluate 16

Review Problems Use the properties of sigma notation and the summation formulas to evaluate the given sums:

Review Problems Find the limit of s(n) as

Review Problems Find the limit: 19

Review Problems LetFind the limit of s(n) as LetFind the limit of s(n) as 4 20

Review Problems Write the definite integral that represents the area of the shaded region? 21

Review Problems Write the definite integral for the area of the region bounded by the graphs of y = 9 - x 2 and y = 0. Write the definite integral for the area of the region lying in the upper half of the ellipse given by 4x 2 + y 2 = 4. 22

Review Problems Sketch the region whose area is indicated by the integral: 23

Review Problems Given that Find 10 Given that Find 6 24

Review Problems Given that Find Given that

Review Problems Consider Find and Consider Find 26

Review Problems Evaluate (2x 2 + 5) 2 Find the average value of y = x 3 over the interval [0,2] 2 27

Review Problems Use the Fundamental Theorem of Calculus to evaluate: 1 28

Review Problems Evaluate 29

Review Problems Evaluate 30

Review Problems Evaluate 31

Review Problems Evaluate 4 32

Review Problems Use the Trapezoidal Rule, with n = 4, to approximate Use the Trapezoidal Rule, with n = 4, to approximate

Review Problems Consider the region bounded by the x-axis, the function x = 1, and x = -1 Use the Trapezoidal Rule, with n = 4, to approximate the area of the region. (Round the answer to three decimal places.)

Answers Upper, 97; lower, y = x 2 - x

Answers graph (2x 2 + 5)