To determine Calculate the value of integral ∫ 0 2 π ∫ 0 1 ∫ r 1 / 2 − r 2 3 r d z d r d θ of cylindrical coordinates. Explanation Calculation: Determine the value of the integral ∫ 0 2 π ∫ 0 1 ∫ r 1 / 2 − r 2 3 r d z d r d θ . I = ∫ 0 2 π ∫ 0 1 ∫ r 1 / 2 − r 2 3 r d z d r d θ = 3 ∫ 0 2 π ∫ 0 1 [ r z ] r 1 / 2 − r 2 d r d θ = 3 ∫ 0 2 π ∫ 0 1 [ r ( 2 − r 2 ) − 1 2 d r − r 2 d r ] d θ = 3 ∫ 0 2 π [ ∫ 0 1 r ( 2 − r 2 ) − 1 2 d r − ∫ 0 1 r 2 d r ] d θ (1) Consider t = 2 − r 2 . Differentiate the term t with respect to r . d t d r = − 2 r − d t 2 = r d r Determine the limits. t = 2 − r 2 Substitute 1 for r . t = 2 − 1 2 = 1 Substitute 0 for r . t = 2 − 0 = 2 Substitute t for 2 − r 2 and − d t 2 for r d r in Equation (1)

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 14.7, Problem 27E

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Calculate the value of integral ∫02π∫01∫r1/2−r23rdzdrdθ of cylindrical coordinates.

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Chapter 14.7, Problem 26E

Chapter 14.7, Problem 28E

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1. Consider the following system of equations: x13x2 + 4x3 - 5x4 = 7 -2x13x2 + x3 - 6x4 = 7 x16x213x3 - 21x4 = 28 a) Solve the system. Write your solution in parametric and vector form. b) What is a geometric description of the solution. 7 c) Is v = 7 in the span of the set S= [28. 1 HE 3 -5 3 ·6 ? If it is, write v 6 as a linear combination of the vectors in S. Justify. d) How many solutions are there to the associated homogeneous system for the system above? Justify. e) Let A be the coefficient matrix from the system above. Find the set of all solutions to Ax = 0. f) Is there a solution to Ax=b for all b in R³? Justify.

4. Suppose that A is made up of 5 column vectors in R³, and suppose that the rank(A)=3. a. How many solutions are there to Ax=0? Justify. b. What is a geometric description for the nullspace(A)? Justify. c. Do the column vectors of A span R³? Justify. d. Is A invertible? Justify.

Chapter 14 Solutions

University Calculus: Early Transcendentals (4th Edition)

Ch. 14.1 - In Exercises 1-14. evaluate the iterated...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - Prob. 10E

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