Question

Hi,

I'm looking at Part I of Problem Set 7 on Double Integrals. It's the supplementary problem 3A-6.

I tried approaching the question by visualising the integrand in 3D and how it cuts region R in the x-y plane. It worked for the first three integrals but the last three threw me off. I checked the answers but I still cannot understand them.

Is there a better way to approach these questions? And if possible, explain how the last three integrals can be solved.

Thank you!

Answer 1

The question link : http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/3.-double-integrals-and-line-integrals-in-the-plane/part-a-double-integrals/problem-set-7/MIT18_02SC_SupProb3.pdf

The answer link:
http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/3.-double-integrals-and-line-integrals-in-the-plane/part-a-double-integrals/problem-set-7/MIT18_02SC_SupProbSol3.pdf

Answer 2

omg ok.

I managed to solve them using the mathlet that plots 3D graphs. The volumes cancel out nicely for: \[\int\limits_{}^{}\int\limits_{}^{} x ^{2}y\ dA\] \[\int\limits_{}^{}\int\limits_{}^{} xy\ dA\]

Then for \[\int\limits_{}^{}\int\limits_{}^{} x ^{2 } +y\ dA\] I realized I could take out y from the integrals which reduces to \[\int\limits_{}^{}\int\limits_{}^{} x ^{2}\ dA\] + \[\int\limits_{}^{}\int\limits_{}^{}y\ dA\] the second integral being zero.

If anyone has a smarter way to solve without using the mathlet please let me know.