91Ó°ÊÓ

Write the given iterated integral as an iterated integral with the order of integration interchanged. Hint: Begin by sketching a region \(S\) and representing it in two ways, as in Example \(5 .\) $$ \int_{0}^{1} \int_{x^{2}}^{x^{1 / 4}} f(x, y) d y d x $$

Let \(S\) be a lamina in the \(x y\) -plane with center of mass at the origin, and let \(L\) be the line \(a x+b y=0\), which goes through the origin. Show that the (signed) distance \(d\) of a point \((x, y)\) from \(L\) is \(d=(a x+b y) / \sqrt{a^{2}+b^{2}}\), and use this to conclude that the moment of \(S\) with respect to \(L\) is \(0 .\) Note: This shows that a lamina will balance on any line through its center of mass.

Evaluate $$ \int_{0}^{1} \int_{0}^{1} x y e^{x^{2}+y^{2}} d y d x $$

Find the volume of the solid trapped between the surface \(z=\cos x \cos y\) and the \(x y\) -plane, where \(-\pi \leq x \leq \pi\) \(-\pi \leq y \leq \pi\).

Write the given iterated integral as an iterated integral with the order of integration interchanged. Hint: Begin by sketching a region \(S\) and representing it in two ways, as in Example \(5 .\) $$ \int_{1 / 2}^{1} \int_{x^{3}}^{x} f(x, y) d y d x $$

Show that $$ \int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{\left(1+x^{2}+y^{2}\right)^{2}} d y d x=\frac{\pi}{4} $$

Write the given iterated integral as an iterated integral with the order of integration interchanged. Hint: Begin by sketching a region \(S\) and representing it in two ways, as in Example \(5 .\) $$ \int_{0}^{1} \int_{-y}^{y} f(x, y) d x d y $$

Evaluate each iterated integral. $$ \int_{-2}^{2} \int_{-1}^{1}\left|x^{2} y^{3}\right| d y d x $$

Recall the formula \(A=\frac{1}{2} r^{2} \theta\) for the area of the sector of a circle of radius \(r\) and central angle \(\theta\) radians (Section \(10.7) .\) Use this to obtain the formula $$ A=\frac{r_{1}+r_{2}}{2}\left(r_{2}-r_{1}\right)\left(\theta_{2}-\theta_{1}\right) $$ for the area of the polar rectangle \(\left\\{(r, \theta): r_{1} \leq r \leq r_{2},\right.\), \(\left.\theta_{1} \leq \theta \leq \theta_{2}\right\\}\).

Write the given iterated integral as an iterated integral with the order of integration interchanged. Hint: Begin by sketching a region \(S\) and representing it in two ways, as in Example \(5 .\) $$ \int_{-1}^{0} \int_{-\sqrt{y+1}}^{\sqrt{y+1}} f(x, y) d x d y $$