Calculus III - Problem Solving Drill 18: Double Integrals in Polar Coordinates and Applications of Double Integrals

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1 Calculus III - Problem Solving Drill 8: Double Integrals in Polar Coordinates and Applications of Double Integrals Question No. of 0 Instructions: () ead the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the. State whether it would be easier to evaluate (, ) f x y da using polar or rectangular coordinates: Question #0 (A) Polar Coordinates. (B) ectangular Coordinates. (C) It is just as easy either way. (D) A different coordinate system is needed. (E) No way to tell. emember that polar coordinate are easier to use when the region involve polar curves. B. Correct! This is the correct answer; rectangular coordinates would be the easiest way to evaluate the integral. (E) Incorrect! Because the region is relatively easy to write down in rectangular coordinates, it would generally be easier to evaluate the integral using rectangular coordinates. apidlearningcenter.com apid Learning Inc. All ights eserved

2 Question No. 2 of 0 Instructions: () ead the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the 2. State whether it would be easier to evaluate (, ) f x y da using polar or rectangular coordinates: Question #02 (A) Polar Coordinates. (B) ectangular Coordinates. (C) It is just as easy either way. (D) A different coordinate system is needed. (E) No way to tell. A. Correct! This is the correct answer; polar coordinates would be the easiest way to evaluate the integral. Because the region is easily described using polar coordinates, it generally would not be easier to evaluate the integral in rectangular coordinates. (E) Incorrect! Because the region is relatively easy to write down in polar coordinates, it would generally be easier to evaluate the integral using polar coordinates. apidlearningcenter.com apid Learning Inc. All ights eserved

3 Question No. 3 of 0 Instructions: () ead the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the 3. Use rectangular coordinates to describe the region: Question #03 (A) = { 2 x,0 y 3} (B) = { x 2,0 y 3} (C) = {0 x 3, y 2} (D) = { r 2,0 θ 3} (E) = {0 x 2, 0 y 2} Look at the x and y values covered by the shaded region. B. Correct! This is the correct answer. x is between - and 2 and y is between 0 and 3. Look at the x and y values covered by the shaded region. Look at the x and y values covered by the shaded region. (E) Incorrect! Look at the x and y values covered by the shaded region. By observing what x and y values are covered by the shaded region, we determine that = { x 2,0 y 3}. apidlearningcenter.com apid Learning Inc. All ights eserved

4 Question No. of 0 Instructions: () ead the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the. Let = { r 3, 0 θ π}. Evaluate x + yda. Question #0 (A) 26/3 (B) 2 (C) 26 (D) 23 π (E) 52/3 A. Incorrect ewrite as a double polar integral. ewrite as a double polar integral. ewrite as a double polar integral. ewrite as a double polar integral. E. Correct! This is the correct answer, found by rewriting as a double polar integral. If = { r 3, 0 θ π}, then we have π 3 x + yda = ( rcosθ + rsinθ) rdrdθ 0 π 26 = ( θ + θ) θ cos sin rd 0 3 2π = sinθ cosθ = 3 apidlearningcenter.com apid Learning Inc. All ights eserved

5 Question No. 5 of 0 Instructions: () ead the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the 5. Let be the region bounded in the first quadrant by the circle x + y = 9. Evaluate xyda. Question #05 (A) 8/ (B) 8 (C) / (D) 8/8 (E) ewrite as a double polar integral. ewrite as a double polar integral. ewrite as a double polar integral. D. Correct! This is the correct answer, found by rewriting as a double polar integral. E. Incorrect! ewrite as a double polar integral. π The region bounded by x + y = 9 can be expressed as = {0 r 3, 0 θ }, then we have 2 π /2 3 xy da = ( r cosθ r sinθ) r drdθ 0 0 π /2 3 = r cosθ sinθ dθ 0 0 π /2 8 = θ θ θ cos sin d 0 π /2 8 2 = cos θ = 8 apidlearningcenter.com apid Learning Inc. All ights eserved

6 Question No. 6 of 0 Instructions: () ead the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the 6. Find the mass of the lamina over = {0 x 5, y } given that the density at (x, y) is ρ = 2 xy. Question #06 (A) 20 (B) 90 (C) 5 (D) 20 (E) 050 emember that mass is found by integrating the density function over. emember that mass is found by integrating the density function over. choice emember that mass is found by integrating the density function over. emember that mass is found by integrating the density function over. E. Correct! This the correct answer, found by integrating the density function over m = ( x, y) da = xy dxdy = 2x y dy = 50y dy = y = We have ρ ( ) apidlearningcenter.com apid Learning Inc. All ights eserved

7 Question No. 7 of 0 Instructions: () ead the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the 7. Find the x coordinate of the center of mass of the lamina over = {0 x 5, y } given that the density at (x, y) is ρ = 2 xy. Question #07 (A) 85 (B) 85/28 (C) 0/3 (D) 050 (E) 28/85 ecall the formula for center of mass. B. Correct! This is the correct answer, found by using the formula for the center of mass. This is the y coordinate of the center of mass. This is the mass of the lamina. E. Incorrect! ecall the formula for center of mass. M My ecall that the formula for center of mass is ( x, y) = x, m m where Mx = y ρ ( x, y) da, (, ) My = xρ( x, y) da, and m = ρ x yda. First, we find m = ρ( x, y) da = xy dxdy = ( 2x y ) dy = 50y dy = y = M = y ( x, y) da = xy dxdy = 2x y dy = 50y dy = y =. Now, Now, x ρ ( ) ( 2 ) 85 the x coordinate of the center of mass is: x = = apidlearningcenter.com apid Learning Inc. All ights eserved

8 Question No. 8 of 0 Instructions: () ead the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the 8. Find the y coordinate of the center of mass of the lamina over = {0 x 5, y } given that the density at (x, y) is ρ = 2 xy. Question #08 (A) 85 (B) 85/28 (C) 0/3 (D) 050 (E) 28/85 ecall the formula for center of mass. This is the x coordinate of the center of mass. C. Correct! This is the correct answer, found by using the formula for the center of mass. This is the mass of the lamina. E. Incorrect! ecall the formula for center of mass. M My ecall that the formula for center of mass is ( x, y) = x, m m where Mx = y ρ ( x, y) da, (, ) My = xρ( x, y) da, and m = ρ x yda. First, we find m = ρ( x, y) da = xy dxdy = ( 2x y ) dy = 50y dy = y = Now, My = xρ( x, y) da = x y dxdy = x y dy = y dy = y = Now, the y coordinate of the center of mass is: y = = apidlearningcenter.com apid Learning Inc. All ights eserved

9 Question No. 9 of 0 Instructions: () ead the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the 9. Let X and Y be random variables with probability density function f( x, y ) =. Find the probability 36 that 0 X 5,3 Y. Question #09 (A) 5/36 (B) /36 (C) 5 (D) /9 (E) A. Correct! This is the correct answer, found by evaluating the double integral. emember that probability is found by evaluating a double integral. A Feedback on Each Answer Probability, by definition, must be between 0 and. emember that probability is found by evaluating a double integral. E. Incorrect! Probability, by definition, must be between 0 and. If f( x, y ) = is the probability density function, then the probability of the event that 36 0 X 5,3 Y is found by evaluating the double integral as follows: dxdy = x dy = dy = y = apidlearningcenter.com apid Learning Inc. All ights eserved

10 Question No. 0 of 0 Instructions: () ead the problem and answer choices carefully (2) Work the problems on paper as needed (3) Pick the 0. Find the area of the surface S of f( x, y) = 25 x y that lies above the = 0 r 3,0 t 2π. region { } Question #0 (A) 2 π (B) 0 (C) 6 π (D) 5 π (E) 0 π emember the formula for surface area. emember the formula for surface area. emember the formula for surface area. emember the formula for surface area. E. Correct! This is the correct answer, found by using the formula for surface area. 5 The formula for surface area is + fx + fy da which yields: + fx + fy da = da. 25 x y Since the region is a circle, it will generally be easier to change to polar coordinates by letting x = r cos t and y = r sint. The integral then becomes: π π r π 2 2π da = drdt = 5 25 r dt = 5dt = 5t = 0π x y r apidlearningcenter.com apid Learning Inc. All ights eserved