Double Integrals
asked 2014-07-17 10:21:11 -0600
So with the problems he put on the board at the end of class today, I worked out the first double integral and was just hoping someone could verify if I'm doing it right.
So he gave us the domain :
and the double integral as: $$\int\int x^2y\delta A$$
for which I came up with the integrals being: $$\int _0 ^2 \int _0 ^{4-x^2} x^2y \delta y \delta x$$ $$=\int _0 ^2 \frac 12x^2y^2\delta x \mid _0 ^{4-x^2}$$ $$=\int _0 ^2 \frac 12 x^2 (4-x^2)^2 \delta x$$ $$= \int _0 ^2 \frac 12 x^2(16-8x^2+x^4)\delta x$$ $$=\int _0 ^2 8x^2 -4x^4 + \frac 12 x^6 \delta x$$ $$=\frac 83 x^3 - \frac 45 x^5 + \frac {1}{14} x^7 \mid _0 ^2$$ $$= \frac 83 (8) - \frac 45(32) + \frac{1}{14}(128)$$ $$= \frac{64}{3} -\frac{128}{5} +\frac{128}{14}$$
This method seems to make sense to me(hoping that my terms of integration are correct), so I'm just hoping that someone else follows this too, or you could explain and easier way to do it. Thanks!
