MATH237: Calculus 3 Written Assignment 3 Fall 2022
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MATH237: Calculus 3
Written Assignment 3
Fall 2022
Problems
Q1. Evaluate the following integrals.
(a) \0 1 \^y(1) sin(x3 )dxdy . [Hint: Identify as a double integral 11D then try another order.] (b) \\D ^x2 + y2 dA, where D = {(x,y): x2 + y2 ≤ 3, −x ≤ y ≤ x}.
\\\
and (0, 0, 1).
Q2. Consider the region D in the first quadrant of R2 bounded by the curves y = x3 , 9y = x3 ,
x = y3 and 16x = y3 , as pictured below.
4
3
2
1
1 2 3 4
(a) Find a transformation (u,v) = T(x,y) such that the image T(D) of D under T is a rectangle in the uv-plane. Justify your answer.
∂(x,y)
(c) Use the change of variables theorem to compute llD 1 dA.
(d) What geometric quantity does the integral in (c) represent?
Q3. Let E be the region in the first octant of R3 that lies inside the sphere x2 +y2 + z2 = 4 and above the cone z = ^x2 + y2 .
(a) Set up (but do not evaluate) iterated integrals that compute the volume of E ...
(i) ...in Cartesian coordinates, with order of integration dxdy dz .
(ii) ...in cylindrical coordinates, with order of integration dz dr dθ .
(iii) ...in spherical coordinates, with order of integration dρdϕdθ .
Be sure to provide some justification for your answers.
(b) Evaluate one of your integrals from part (a). [To check your work, you may want to evaluate them all. But only submit one evaluation. The TAs will only grade one.]
2022-12-06