91Ó°ÊÓ

Evaluate the iterated integrals. $$ \int_{0}^{1} \int_{2 \sqrt{x}}^{2 \sqrt{x}+1}(x y+1) d y d x $$

Evaluate the iterated integrals. $$ \int_{e}^{2} \int_{\ln u}^{2}(v+\ln u) d v d u $$

Evaluate the iterated integrals. $$ \int_{1}^{2} \int_{-u^{2}-1}^{-u}(8 u v) d v d u $$

Evaluate the iterated integrals. $$ \int_{0}^{1} \int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}}\left(2 x+4 x^{3}\right) d x d y $$

Evaluate the iterated integrals. $$ \int_{0}^{1 / 2} \int_{-\sqrt{1-4 y^{2}}}^{\sqrt{1-4 y^{2}}} 4 d x d y $$

Let \(D\) be the region bounded by \(y=1-x^{2}, y=4-x^{2},\) and the \(x\) - and \(y\) -axes. a. Show that $$ \iint_{D} x d A=\int_{0}^{1} \int_{1-x^{2}}^{4-x^{2}} x d y d x+\int_{1}^{2} \int_{0}^{4-x^{2}} x d y d x \text { by } $$ dividing the region \(D\) into two regions of Type I. b. Evaluate the integral \(\iint_{D} x d A\).

a. Show that $$ \iint_{D} y^{2} d A=\int_{-1}^{0} \int_{-x}^{2-x^{2}} y^{2} d y d x+\int_{0}^{1} \int_{x}^{2-x^{2}} y^{2} d y d x $$ by dividing the region \(D\) into two regions of Type I, where \(D=\left\\{(x, y) \mid y \geq x, y \geq-x, y \leq 2-x^{2}\right\\}\) b. Evaluate the integral \(\iint_{D} y^{2} d A\).

Let \(D\) be the region bounded by \(y=x^{2}, y=x+2\), and \(y=-x\). a. Show that $$ \iint_{D} x d A=\int_{0}^{1} \int_{-y}^{\sqrt{y}} x d x d y+\int_{1}^{2} \int_{y-2}^{\sqrt{y}} x d x d y $$ by dividing the region \(D\) into two regions of Type II, where \(D=\left\\{(x, y) \mid y \geq x^{2}, y \geq-x, y \leq x+2\right\\}\). b. Evaluate the integral \(\iint_{D} x d A\).

The region \(D\) bounded by \(y=\cos x, y=4 \cos x\), and \(x=\pm \frac{\pi}{3}\) is shown in the following figure. Find the area \(A(D)\) of the region \(D\).

Find the area \(A(D)\) of the region \(D=\left\\{(x, y) \mid y \geq 1-x^{2}, y \leq 4-x^{2}, y \geq 0, x \geq 0\right\\}\)