Calculus 3

Math 3C Quiz 4 Sections 15.1 – 15.8

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1) (a) Evaluate the iterated integral ∫ ∫ 𝑥 + 2𝑥𝑦 + 𝑦 𝑑𝑦 𝑑𝑥!

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(b) Evaluate ∬ 𝑑𝐴% , with the region R bounded by: 0 ≤ 𝑥 ≤ 𝑦& , 1 ≤ 𝑦 ≤ 9

2) For the given double integral, sketch the region of integration, then write an equivalent double integral with the order of integration reversed. Do not attempt to evaluate (because the function is unknown).

/ / 𝑓(𝑥, 𝑦) 𝑑𝑥 𝑑𝑦 √(

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3) As of April 20, 2020, Symbolab could not compute ∫ ∫ 1 + 𝑦& 𝑑𝑦 𝑑𝑥 *+$,-!

" . " . Change the given integral into an

equivalent polar integral. Symbolab was able to evaluate the polar version of this integral (the result is 6885π/16). However, you do not need to evaluate either integral - not even for extra credit - since a computer can do it much faster, and possibly more accurately, than you or your teacher. It would be very helpful to sketch the region of integration before changing the integral to polar form.

4) Set up two integrals that would calculate the volume of the region in the 1st octant between the plane 3x+2y+z=6 and the x-y plane. Then calculate one of them. (Recall the 1st octant is where x, y, and z ³ 0) (i) ∫ ∫ ∫ 𝑑𝑧 𝑑𝑦 𝑑𝑥 (ii) ∫ ∫ ∫ 𝑑𝑦 𝑑𝑥 𝑑𝑧

5) Below is a graph of the cylindrical coordinates equation 𝑟 = sin(𝑧) + 2, for ,!/ # < 𝑧 < 2𝜋. Set up only an integral

that would give the volume of this beautiful, expensive vase.

6) Refer to the handout entitled A Story on Changing Variables. Follow Gertrude’s example to find a change of variables that will allow you to express the area of the parallelogram with a single iterated integral. Show your calculation of the Jacobian then set up the iterated integral with the new coordinates. The parallelogram is bounded by the lines: y = x + 8; y = x – 2; x + y = 3; x + y = 8