Question

Match the integrals with the type of coordinates which make them the easiest to do. Put the letter of the coordinate system to the left of the number of the integral.

1. ∫(0 to 1)∫(0 to y^2) 1/x dx dy
2. ∫∫∫E dV where E is: x^2+y^2+z^2≤4, x≥0, y≥0, z≥0
3. ∫∫∫E z^2 dV where E is: −2≤z≤2, 1≤x^2+y^2≤2
4. ∫∫∫E z dV where E is: 1≤x≤2, 3≤y≤4, 5≤z≤6
5. ∫∫D 1/(x^2+y^2) dA where D is: x^2+y^2≤4

A. cylindrical coordinates
B. cartesian coordinates
C. polar coordinates
D. spherical coordinates

Answer 1

Let's choose the coordinate system for each integral while trying to justify why :

1. B, cartesian. Why ? Because the integration bounds are polynomial expressions of x or y and so there will be no problem in integrating this (except that the integral of1/x dx from 0 to y^2 is not well defined).
2. D, spherical. Why ? Because the domain is a 1/8 th of a sphere, so going to spherical will make the equation of this domain much easier.
3. A, cylindrical. Why ? Because the expression in x and y is $x^2+y^2$ and the expression in z is linear.
4. B, cartesian. For the same reasons as in (1), domain's bounds expressions are quite simple in cartesian coordinates, so there is no need to change them.
5. C, polar. We encounter once again an expression of the form $x^2+y^2$, but compared to (3) we are in a 2-dimensional space, so we chose the polar coordiantes.