8. Rewrite the integral $\int_{0}^{3} \int_{\sqrt{9 y^{2}}}^{\sqrt{9 y^{2}}} 11 d x d y$ to change the order of integration, and then evaluate it. |Tip: You may use a geometric argument for evaluation, if you find one.
Solution
Step 1 ：First, we notice that the limits of the inner integral are the same, which means the integral over that range is zero. Therefore, the double integral is also zero.
Step 2 ：To change the order of integration, we need to express the limits in terms of x. However, since the integral is zero, changing the order of integration will not change the result.
Step 3 ：So, the integral $\int_{0}^{3} \int_{\sqrt{9 y^{2}}}^{\sqrt{9 y^{2}}} 11 d x d y$ is equal to zero.
Step 4 ：We can check our result by noting that the integrand is a constant, so the integral should be equal to the area of the region of integration times the value of the integrand. But the region of integration has zero area, so the integral is zero.
Step 5 ：Therefore, the final answer is \(\boxed{0}\)
