91Ó°ÊÓ

Evaluate \(\iint_{D} x y d A\) if \(D\) is the triangular region with vertices (0,0),(1,1),(2,0) .

For each of the triple integrals,(a) sketch the domain of integration in \(\mathbb{R}^{3},(\mathbf{b})\) write an equivalent expression in which the first integration is with respect to \(x,\) and \((\mathbf{c})\) write an equivalent expression in which the first integration is with respect to \(y\). $$ \int_{-1}^{1}\left(\int_{0}^{\sqrt{1-x^{2}}}\left(\int_{0}^{1} f(x, y, z) d z\right) d y\right) d x $$

For each of the triple integrals,(a) sketch the domain of integration in \(\mathbb{R}^{3},(\mathbf{b})\) write an equivalent expression in which the first integration is with respect to \(x,\) and \((\mathbf{c})\) write an equivalent expression in which the first integration is with respect to \(y\). $$ \int_{0}^{1}\left(\int_{x}^{1}\left(\int_{0}^{x} f(x, y, z) d z\right) d y\right) d x $$

(a) sketch the domain of integration \(D\) in the \(x y\) -plane and (b) write an equivalent expression with the order of integration reversed. $$ \int_{0}^{1}\left(\int_{e^{x}}^{e} f(x, y) d y\right) d x $$

For each of the triple integrals,(a) sketch the domain of integration in \(\mathbb{R}^{3},(\mathbf{b})\) write an equivalent expression in which the first integration is with respect to \(x,\) and \((\mathbf{c})\) write an equivalent expression in which the first integration is with respect to \(y\). $$ \int_{0}^{1}\left(\int_{x}^{1}\left(\int_{0}^{y} f(x, y, z) d z\right) d y\right) d x $$

(a) sketch the domain of integration \(D\) in the \(x y\) -plane and (b) write an equivalent expression with the order of integration reversed. $$ \int_{0}^{1}\left(\int_{y^{2}}^{y} f(x, y) d x\right) d y $$

(a) sketch the domain of integration \(D\) in the \(x y\) -plane and (b) write an equivalent expression with the order of integration reversed. $$ \int_{0}^{\pi / 2}\left(\int_{0}^{\sin y} f(x, y) d x\right) d y $$

(a) sketch the domain of integration \(D\) in the \(x y\) -plane and (b) write an equivalent expression with the order of integration reversed. $$ \int_{-1}^{1}\left(\int_{0}^{\sqrt{1-x^{2}}} f(x, y) d y\right) d x $$

(a) sketch the domain of integration \(D\) in the \(x y\) -plane and (b) write an equivalent expression with the order of integration reversed. $$ \int_{-3}^{6}\left(\int_{\frac{1}{2} y}^{-1+\sqrt{3+y}} f(x, y) d x\right) d y $$

Consider the iterated integral \(\int_{\frac{1}{4}}^{1}\left(\int_{\frac{1}{x}}^{4} y e^{x y} d y\right) d x\). (a) Sketch the domain of integration \(D\) in the \(x y\) -plane. (b) Write an equivalent expression with the order of integration reversed. (c) Evaluate the integral using whatever method seems best.