Re: 232.q1.5e dbl trig int
First, let's plot $R$...
View attachment 7266
Let's try vertical strips:
$$V=\int_{-\sqrt{5}}^{\sqrt{5}}\int_{-\sqrt{5-x^2}}^{\sqrt{5-x^2}} 3x^5-y^2\sin(y)+5\,dy\,dx$$
One thing that makes our life a great deal easier is to recognize that the following term in the integrand:
$$y^2\sin(y)$$
is an odd function, and since the limits are symmetrical about the $y$-axis, it goes to zero, so we may state:
$$V=\int_{-\sqrt{5}}^{\sqrt{5}}\int_{-\sqrt{5-x^2}}^{\sqrt{5-x^2}} 3x^5+5\,dy\,dx$$
$$V=\int_{-\sqrt{5}}^{\sqrt{5}}3x^5+5\int_{-\sqrt{5-x^2}}^{\sqrt{5-x^2}}\,dy\,dx$$
$$V=2\int_{-\sqrt{5}}^{\sqrt{5}} \sqrt{5-x^2}\left(3x^5+5\right)\,dx$$
Again, we may use the odd-function rule to simplify this to:
$$V=10\int_{-\sqrt{5}}^{\sqrt{5}} \sqrt{5-x^2}\,dx$$
Now, we may use the even function rule to state:
$$V=20\int_{0}^{\sqrt{5}} \sqrt{5-x^2}\,dx$$
I will let you take it from here. :)
Can you set this up using horizontal strips? How about polar coordinates?
