Chapter 14

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

Let \(R\) be the region in the first quadrant of the \(x y\) -plane bounded by the hyperbolas \(x y=1, x y=9\) and the lines \(y=x, y=4 x\) Use the transformation \(x=u / v, y=u v\) with \(u>0\) and \(v>0\) to rewrite $$ \iint_{R}(\sqrt{\frac{y}{x}}+\sqrt{x y}) d x d y $$ as an integral over an appropriate region \(G\) in the \(u v\) -plane. Then evaluate the \(u \boldsymbol{v}\) -integral over \(\boldsymbol{G}\)

The integrals we have seen so far suggest that there are preferred orders of integration for cylindrical coordinates, but other orders usually work well and are occasionally easier to evaluate. Evaluate the integrals. $$\int_{0}^{1} \int_{0}^{\sqrt{z}} \int_{0}^{2 \pi}\left(r^{2} \cos ^{2} \theta+z^{2}\right) r d \theta d r d z$$

Evaluate the integrals in Exercises \(7-20\). $$\int_{1}^{e} \int_{1}^{e^{2}} \int_{1}^{e^{3}} \frac{1}{x y z} d x d y d z$$

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{-1}^{1} \int_{0}^{\sqrt{1-x^{2}}} d y d x$$

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The lines \(y=x, y=x / 3,\) and \(y=2\).

Evaluate the iterated integral. $$\int_{0}^{\ln 2} \int_{1}^{\ln 5} e^{2 x+y} d y d x$$

The integrals we have seen so far suggest that there are preferred orders of integration for cylindrical coordinates, but other orders usually work well and are occasionally easier to evaluate. Evaluate the integrals. $$\int_{0}^{2} \int_{r-2}^{\sqrt{4-r^{2}}} \int_{0}^{2 \pi}(r \sin \theta+1) r d \theta d z d r$$

Find the first moment about the \(y\) -axis of a thin plate of density \(\delta(x, y)=1\) covering the infinite region under the curve \(y=e^{-x^{2} / 2}\) in the first quadrant.

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{1} \int_{0}^{3-3 x} \int_{0}^{3-3 x-y} d z d y d x$$

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{0}^{1} \int_{0}^{\sqrt{1-y^{2}}}\left(x^{2}+y^{2}\right) d x d y$$

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The lines \(y=1-x\) and \(y=2\) and the curve \(y=e^{x}\).

Evaluate the iterated integral. $$\int_{0}^{1} \int_{1}^{2} x y e^{x} d y d x$$

Let \(D\) be the region bounded below by the plane \(z=0,\) above by the sphere \(x^{2}+y^{2}+z^{2}=4,\) and on the sides by the cylinder \(x^{2}+y^{2}=1 .\) Set up the triple integrals in cylindrical coordinates that give the volume of \(D\) using the following orders of integration. a. \(d z d r d \theta\) b. \(d r d z d \theta\) c. \(d \theta d z d r\)

Polar moment of inertia of an elliptical plate \(\quad\) A thin plate of constant density covers the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1, a>0, b>0,\) in the \(x y\) -plane. Find the first moment of the plate about the origin. (Hint: Use the transformation \(x=a r \cos \theta, y=b r \sin \theta .\)

Find the moment of inertia about the \(x\) -axis of a thin plate bounded by the parabola \(x=y-y^{2}\) and the line \(x+y=0\) if \(\delta(x, y)=x+y\).

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{\pi / 6} \int_{0}^{1} \int_{-2}^{3} y \sin z d x d y d z$$

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{0}^{2} \int_{0}^{\sqrt{4-y^{2}}}\left(x^{2}+y^{2}\right) d x d y$$

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The lines \(y=2 x, y=x / 2,\) and \(y=3-x\)

Evaluate the iterated integral. $$\int_{-1}^{2} \int_{0}^{\pi / 2} y \sin x d x d y$$

Let \(D\) be the region bounded below by the cone \(z=\sqrt{x^{2}+y^{2}}\) and above by the paraboloid \(z=2-x^{2}-y^{2} .\) Set up the triple integrals in cylindrical coordinates that give the volume of \(D\) using the following orders of integration. a. \(d z d r d \theta\) b. \(d r d z d \theta\) c. \(d \theta d z d r\)

The area of an ellipse The area \(\pi a b\) of the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) can be found by integrating the function \(f(x, y)=1\) over the region bounded by the ellipse in the \(x y-\) plane. Evaluating the integral directly requires a trigonometric substitution. An easier way to evaluate the integral is to use the transformation \(x=a u, y=b v\) and evaluate the transformed integral over the disk \(G: u^{2}+v^{2} \leq 1\) in the \(u v\) -plane. Find the area this way.

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{-a}^{a} \int_{-\sqrt{a^{2}-x^{2}}}^{\sqrt{a^{2}-x^{2}}} d y d x$$

Find the mass of a thin plate occupying the smaller region cut from the ellipse \(x^{2}+4 y^{2}=12\) by the parabola \(x=4 y^{2}\) if \(\delta(x, y)=5 x\).

Evaluate the integrals in Exercises \(7-20\). $$\int_{-1}^{1} \int_{0}^{1} \int_{0}^{2}(x+y+z) d y d x d z$$

Evaluate the iterated integral. $$\int_{\pi}^{2 \pi} \int_{0}^{\pi}(\sin x+\cos y) d x d y$$

Give the limits of integration for evaluating the integral \(\iiint f(r, \theta, z) d z \, r \, d r d \theta\) as an iterated integral over the region that is bounded below by the plane \(z=0,\) on the side by the cylinder \(r=\cos \theta,\) and on top by the paraboloid \(z=3 r^{2}\).

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} \int_{0}^{\sqrt{9-x^{2}}} d z d y d x$$

Find the center of mass of a thin triangular plate bounded by the \(y\) -axis and the lines \(y=x\) and \(y=2-x\) if \(\delta(x, y)=6 x+3 y+3\).

The integrals and sums of integrals in Exercises \(13-18\) give the areas of regions in the \(x y\) -plane. Sketch each region, label each bounding curve with its equation, and give the coordinates of the points where the curves intersect. Then find the area of the region. $$\int_{0}^{6} \int_{y^{2} / 3}^{2 y} d x d y$$

Write an iterated integral for \(\iint_{R} d A\) over the described region \(R\) using (a) vertical cross-sections, (b) horizontal cross-sections. Bounded by \(y=\sqrt{x}, y=0,\) and \(x=9\)

Evaluate the iterated integral. $$\int_{1}^{4} \int_{1}^{e} \frac{\ln x}{x y} d x d y

Convert the integral $$\int_{-1}^{1} \int_{0}^{\sqrt{1-y^{2}}} \int_{0}^{x}\left(x^{2}+y^{2}\right) d z d x d y$$ to an equivalent integral in cylindrical coordinates and evaluate the result.

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{2} \int_{-\sqrt{4-y^{2}}}^{\sqrt{4-y^{2}}} \int_{0}^{2 x+y} d z d x d y$$

Find the center of mass and moment of inertia about the \(x\) -axis of a thin plate bounded by the curves \(x=y^{2}\) and \(x=2 y-y^{2}\) if the density at the point \((x, y)\) is \(\delta(x, y)=y+1\).

The integrals and sums of integrals in Exercises \(13-18\) give the areas of regions in the \(x y\) -plane. Sketch each region, label each bounding curve with its equation, and give the coordinates of the points where the curves intersect. Then find the area of the region. $$\int_{0}^{3} \int_{-x}^{\sqrt{(2-x)}} d y d x$$

Write an iterated integral for \(\iint_{R} d A\) over the described region \(R\) using (a) vertical cross-sections, (b) horizontal cross-sections. Bounded by \(y=\tan x, x=0,\) and \(y=1\)

Evaluate the iterated integral. $$\int_{-1}^{2} \int_{1}^{2} x \ln y d y d x$$

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{1} \int_{0}^{2-x} \int_{0}^{2-x-y} d z d y d x$$

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{1}^{\sqrt{3}} \int_{1}^{x} d y d x$$

Find the center of mass and the moment of inertia about the \(y\) -axis of a thin rectangular plate cut from the first quadrant by the lines \(x=6\) and \(y=1\) if \(\delta(x, y)=x+y+1\).

The integrals and sums of integrals in Exercises \(13-18\) give the areas of regions in the \(x y\) -plane. Sketch each region, label each bounding curve with its equation, and give the coordinates of the points where the curves intersect. Then find the area of the region. $$\int_{0}^{\pi / 4} \int_{\sin x}^{\cos x} d y d x$$

Write an iterated integral for \(\iint_{R} d A\) over the described region \(R\) using (a) vertical cross-sections, (b) horizontal cross-sections. Bounded by \(y=e^{-x}, y=1,\) and \(x=\ln 3\)

Evaluate the double integral over the given region \(R\). $$\iint_{R}\left(6 y^{2}-2 x\right) d A, \quad R: \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2$$

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{\sqrt{2}}^{2} \int_{\sqrt{4-y^{2}}}^{y} d x d y$$

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{1} \int_{0}^{1-x^{2}} \int_{3}^{4-x^{2}-y} x d z d y d x$$

Find the center of mass and the moment of inertia about the \(y\) -axis of a thin plate bounded by the line \(y=1\) and the parabola \(y=x^{2}\) if the density is \(\delta(x, y)=y+1\).

The integrals and sums of integrals in Exercises \(13-18\) give the areas of regions in the \(x y\) -plane. Sketch each region, label each bounding curve with its equation, and give the coordinates of the points where the curves intersect. Then find the area of the region. $$\int_{-1}^{2} \int_{y^{2}}^{y+2} d x d y$$

Evaluate the double integral over the given region \(R\). $$\iint_{R}\left(\frac{\sqrt{x}}{y^{2}}\right) d A, \quad R: \quad 0 \leq x \leq 4, \quad 1 \leq y \leq 2$$

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{-1}^{0} \int_{-\sqrt{1-x^{2}}}^{0} \frac{2}{1+\sqrt{x^{2}+y^{2}}} d y d x$$