Chapter 15

Sand pile: double and triple integrals The base of a sand pile covers the region in the \(x y\) -plane that is bounded by the parabola \(x^{2}+y=6\) and the line \(y=x\) . The height of the sand above the point \((x, y)\) is \(x^{2} .\) Express the volume of sand as (a) a double integral, (b) a triple integral. Then (c) find the volume.

Evaluate the cylindrical coordinate integrals. \(\int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{\sqrt{2-r^{2}}} d z r d r d \theta\)

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

In Exercises \(1-10,\) sketch the region of integration and evaluate the integral. $$ \int_{0}^{3} \int_{0}^{2}\left(4-y^{2}\right) d y d x $$

In Exercises 1–8, 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 coordinate axes and the line \(x+y=2\)

Water in a hemispherical bowl A hemispherical bowl of radius 5 \(\mathrm{cm}\) is filled with water to within 3 \(\mathrm{cm}\) of the top. Find the volume of water in the bowl.

Volume of rectangular solid Write six different iterated triple integrals for the volume of the rectangular solid in the first octant bounded by the coordinate planes and the planes \(x=1, y=2\) and \(z=3 .\) Evaluate one of the integrals.

Evaluate the cylindrical coordinate integrals. \(\int_{0}^{2 \pi} \int_{0}^{3} \int_{r^{2} / 3}^{\sqrt{18-r^{2}}} d z r d r d \theta\)

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

In Exercises \(1-10,\) sketch the region of integration and evaluate the integral. $$ \int_{0}^{3} \int_{-2}^{0}\left(x^{2} y-2 x y\right) d y d x $$

In Exercises 1–8, 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 \(x=0, y=2 x,\) and \(y=4\)

Solid cylindrical region between two planes Find the volume of the portion of the solid cylinder \(x^{2}+y^{2} \leq 1\) that lies between the planes \(z=0\) and \(x+y+z=2\)

Volume of tetrahedron Write six different iterated triple integrals for the volume of the tetrahedron cut from the first octant by the plane \(6 x+3 y+2 z=6\) . Evaluate one of the integrals.

Evaluate the cylindrical coordinate integrals. \(\int_{0}^{2 \pi} \int_{0}^{\theta / 2 \pi} \int_{0}^{3+24 r^{2}} d z r d r d \theta\)

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

In Exercises \(1-10,\) sketch the region of integration and evaluate the integral. $$ \int_{-1}^{0} \int_{-1}^{1}(x+y+1) d x d y $$

In Exercises 1–8, 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 parabola \(x=-y^{2}\) and the line \(y=x+2\)

Sphere and paraboloid Find the volume of the region bounded above by the sphere \(x^{2}+y^{2}+z^{2}=2\) and below by the paraboloid \(z=x^{2}+y^{2}\)

Volume of solid Write six different iterated triple integrals for the volume of the region in the first octant enclosed by the cylinder \(x^{2}+z^{2}=4\) and the plane \(y=3 .\) Evaluate one of the integrals.

Evaluate the cylindrical coordinate integrals. \(\int_{0}^{\pi} \int_{0}^{\theta / \pi} \int_{-\sqrt{4-r^{2}}}^{3 \sqrt{4-r^{2}}} z d z r d r d \theta\)

The solids in Exercises \(1-12\) all have constant density \(\delta=1\) a. Centroid and moments of inertia Find the centroid and the moments of inertia \(I_{x}, I_{y},\) and \(I_{z}\) of the tetrahedron whose vertices are the points \((0,0,0),(1,0,0),(0,1,0),\) and \((0,0,1) .\) b. Radius of gyration Find the radius of gyration of the tetrahedron about the \(x\) -axis. Compare it with the distance from the centroid to the \(x\) -axis.

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

In Exercises \(1-10,\) sketch the region of integration and evaluate the integral. $$ \int_{\pi}^{2 \pi} \int_{0}^{\pi}(\sin x+\cos y) d x d y $$

In Exercises 1–8, 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 parabola \(x=y-y^{2}\) and the line \(y=-x\)

Two paraboloids Find the volume of the region bounded above by the paraboloid \(z=3-x^{2}-y^{2}\) and below by the paraboloid \(z=2 x^{2}+2 y^{2} .\)

Volume enclosed by paraboloids Let \(D\) be the region bounded by the paraboloids \(z=8-x^{2}-y^{2}\) and \(z=x^{2}+y^{2} .\) Write six different triple iterated integrals for the volume of \(D .\) Evaluate one of the integrals.

Evaluate the cylindrical coordinate integrals. \(\int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{1 / \sqrt{2}-r^{2}} 3 d z r d r d \theta\)

The solids in Exercises \(1-12\) all have constant density \(\delta=1\) Center of mass and moments of inertia A solid "trough" of constant density is bounded below by the surface \(z=4 y^{2}\) , above by the plane \(z=4,\) and on the ends by the planes \(x=1\) and \(x=-1\) . Find the center of mass and the moments of inertia with respect to the three axes.

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

In Exercises \(1-10,\) sketch the region of integration and evaluate the integral. $$ \int_{0}^{\pi} \int_{0}^{x} x \sin y d y d x $$

In Exercises 1–8, 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 curve \(y=e^{x}\) and the lines \(y=0, x=0,\) and \(x=\ln 2\)

The solids in Exercises \(1-12\) all have constant density \(\delta=1\) Center of mass A solid of constant density is bounded below by the plane \(z=0,\) on the sides by the elliptical cylinder \(x^{2}+4 y^{2}=4,\) and above by the plane \(z=2-x\) (see the accompanying figure). a. Find \(\overline{x}\) and \(\overline{y}\) . b. Evaluate the integral $$ M_{x y}=\int_{-2}^{2} \int_{-(1 / 2) \sqrt{4-x^{2}}}^{(1 / 2) \sqrt{4-x^{2}}} \int_{0}^{2-x} z d z d y d x $$ using integral tables to carry out the final integration with respect to \(x\) . Then divide \(M_{x y}\) by \(M\) to verify that \(\overline{z}=5 / 4\) .

Volume inside paraboloid beneath a plane Let \(D\) be the region bounded by the paraboloid \(z=x^{2}+y^{2}\) and the plane \(z=2 y .\) Write triple iterated integrals in the order \(d z d x d y\) and \(d z d y d x\) that give the volume of \(D .\) Do not evaluate either integral.

Evaluate the cylindrical coordinate integrals. \(\int_{0}^{2 \pi} \int_{0}^{1} \int_{-1 / 2}^{1 / 2}\left(r^{2} \sin ^{2} \theta+z^{2}\right) d z r d r d \theta\)

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

In Exercises \(1-10,\) sketch the region of integration and evaluate the integral. $$ \int_{0}^{\pi} \int_{0}^{\sin x} y d y d x $$

In Exercises 1–8, 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 curves \(y=\ln x\) and \(y=2 \ln x\) and the line \(x=e,\) in the first quadrant

Hole in sphere \(A\) circular cylindrical hole is bored through a solid sphere, the axis of the hole being a diameter of the sphere. The volume of the remaining solid is $$ V=2 \int_{0}^{2 \pi} \int_{0}^{\sqrt{3}} \int_{1}^{\sqrt{4-z^{2}}} r d r d z d \theta $$ a. Find the radius of the hole and the radius of the sphere. b. Evaluate the integral.

The solids in Exercises \(1-12\) all have constant density \(\delta=1\) a. Center of mass Find the center of mass of a solid of constant density bounded below by the paraboloid \(z=x^{2}+y^{2}\) and above by the plane \(z=4\) . b. Find the plane \(z=c\) that divides the solid into two parts of equal volume. This plane does not pass through the center of mass.

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

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. \(\int_{0}^{2 \pi} \int_{0}^{3} \int_{0}^{z / 3} r^{3} d r d z d \theta\)

In Exercises \(1-10,\) sketch the region of integration and evaluate the integral. $$ \int_{1}^{\ln 8} \int_{0}^{\ln y} e^{x+y} d x d y $$

In Exercises 1–8, 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 parabolas \(x=y^{2}\) and \(x=2 y-y^{2}\)

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. \(\int_{0}^{6} \int_{0}^{y} x d x d y\)

The solids in Exercises \(1-12\) all have constant density \(\delta=1\) Moments and radii of gyration A solid cube, 2 units on a side, is bounded by the planes \(x=\pm 1, z=\pm 1, y=3,\) and \(y=5\) . Find the center of mass and the moments of inertia and radii of gyration about the coordinate axes.

Evaluate the integrals in Exercises \(7-20\). $$ \int_{0}^{\sqrt{2}} \int_{0}^{3 y} \int_{x^{2}+3 y^{2}}^{8-x^{2}-y^{2}} d z d x d y $$

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. \(\int_{-1}^{1} \int_{0}^{2 \pi} \int_{0}^{1+\cos \theta} 4 r d r d \theta d z\)

In Exercises \(1-10,\) sketch the region of integration and evaluate the integral. $$ \int_{1}^{2} \int_{y}^{y^{2}} d x d y $$

In Exercises 1–8, 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 parabolas \(x=y^{2}-1\) and \(x=2 y^{2}-2\)

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. \(\int_{0}^{2} \int_{0}^{x} y d y d x\)