Chapter 14

Find the volume of the region bounded above by the paraboloid \(z=x^{2}+y^{2} \quad\) and below by the square \(\quad R:-1 \leq x \leq 1\) \(-1 \leq y \leq 1\)

Sketch the region of integration and convert each polar integral or sum of integrals to a Cartesian integral or sum of integrals. Do not evaluate the integrals. $$\int_{0}^{\tan ^{-1} \frac{4}{3}} \int_{0}^{3 \sec \theta} r^{7} d r d \theta+\int_{\tan ^{-1} \frac{4}{3}} \int_{0}^{\pi / 2} r^{7} d r d \theta$$

Evaluate the spherical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{\sec \phi}(\rho \cos \phi) \rho^{2} \sin \phi d \rho d \phi d \theta$$

Cylindrical shells In Section \(6.2,\) we learned how to find the volume of a solid of revolution using the shell method; namely, if the region between the curve \(y=f(x)\) and the \(x\) -axis from \(a\) to \(b\) \((0

Regional population If \(f(x, y)=100(y+1)\) represents the population density of a planar region on Earth, where \(x\) and \(y\) are measured in miles, find the number of people in the region bounded by the curves \(x=y^{2}\) and \(x=2 y-y^{2}\).

Integrate \(f\) over the given region. Triangle \(\quad f(x, y)=x^{2}+y^{2}\) over the triangular region with vertices \((0,0),(1,0),\) and (0,1)

Find the volume of the region bounded above by the ellipitical paraboloid \(z=16-x^{2}-y^{2} \quad\) and \(\quad\) below \(\quad\) by \(\quad\) the \(\quad\) square \(R: 0 \leq x \leq 2,0 \leq y \leq 2\)

The tetrahedron in the first octant bounded by the coordinate planes and the planes passing through \((1,0,0),(0,2,0),\) and (0,0,3) (GRAPH CAN'T COPY).

The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders give the same value and are occasionally easier to evaluate. Evaluate the integrals. $$\int_{0}^{2} \int_{-\pi}^{0} \int_{\pi / 4}^{\pi / 2} \rho^{3} \sin 2 \phi d \phi d \theta d \rho$$

Integrate \(f\) over the given region. Triangle \(\quad f(u, v)=v-\sqrt{u}\) over the triangular region cut from the first quadrant of the \(u v\) -plane by the line \(u+v=1\)

Average temperature in Texas According to the Texas Almanac, Texas has 254 counties and a National Weather Service station in each county. Assume that at time \(t_{0},\) each of the 254 weather stations recorded the local temperature. Find a formula that would give a reasonable approximation of the average temperature in Texas at time \(t_{0} .\) Your answer should involve information that you would expect to be readily available in the Texas Almanac.

Find the volume of the region bounded above by the plane \(z=2-x-y\) and below by the square \(R: 0 \leq x \leq 1\) \(0 \leq y \leq 1\)

The region in the first octant bounded by the coordinate planes, the plane \(y=1-x,\) and the surface \(z=\cos (\pi x / 2), 0 \leq x \leq 1\). (GRAPH CAN'T COPY).

Find the area of the region that lies inside the cardioid \(r=1+\cos \theta\) and outside the circle \(r=1.\)

The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders give the same value and are occasionally easier to evaluate. Evaluate the integrals. $$\int_{\pi / 6}^{\pi / 3} \int_{\csc \phi}^{2 \csc \phi} \int_{0}^{2 \pi} \rho^{2} \sin \phi d \theta d \rho d \phi$$

Integrate \(f\) over the given region. Curved region \(f(s, t)=e^{s}\) Int over the region in the first quadrant of the st-plane that lies above the curve \(s=\ln t\) from \(t=1\) to \(t=2\)

Find the volume of the region bounded above by the plane \(z=y / 2\) and below by the rectangle \(R: 0 \leq x \leq 4,0 \leq y \leq 2\)

Find the area enclosed by one leaf of the rose \(r=12 \cos 3 \theta\).

The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders give the same value and are occasionally easier to evaluate. Evaluate the integrals. $$\int_{0}^{1} \int_{0}^{\pi} \int_{0}^{\pi / 4} 12 \rho \sin ^{3} \phi d \phi d \theta d \rho$$

Find. a. the mass of the solid. b. the center of mass. A solid region in the first octant is bounded by the coordinate planes and the plane \(x+y+z=2 .\) The density of the solid is \(\delta(x, y, z)=2 x\).

Suppose \(f(x, y)\) is continuous over a region \(R\) in the plane and that the area \(A(R)\) of the region is defined. If there are constants \(m\) and \(M\) such that \(m \leq f(x, y) \leq M\) for all \((x, y) \in R,\) prove that $$ m A(R) \leq \iint_{k} f(x, y) d A \leq M A(R) $$

Find the volume of the region bounded above by the surface \(z=2 \sin x \cos y\) and below by the rectangle \(R: 0 \leq x \leq \pi / 2\) \(0 \leq y \leq \pi / 4\).

Find the area of the region enclosed by the positive \(x\) -axis and spiral \(r=4 \theta / 3,0 \leq \theta \leq 2 \pi .\) The region looks like a snail shell.

The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders give the same value and are occasionally easier to evaluate. Evaluate the integrals. $$\int_{\pi / 6}^{\pi / 2} \int_{-\pi / 2}^{\pi / 2} \int_{\csc \phi}^{2} 5 \rho^{4} \sin ^{3} \phi d \rho d \theta d \phi$$

Each gives an integral over a region in a Cartesian coordinate plane. Sketch the region and evaluate the integral. $$\int_{0}^{1} \int_{0}^{\sqrt{1-s^{2}}} 8 t d t d s \quad(\text { the } s t-\text { plane })$$

Suppose \(f(x, y)\) is continuous and nonnegative over a region \(R\) in the plane with a defined area \(A(R) .\) If \(\iint_{R} f(x, y) d A=0,\) prove that \(f(x, y)=0\) at every point \((x, y) \in R\)

Find the volume of the region bounded above by the surface \(z=4-y^{2}\) and below by the rectangle \(R: 0 \leq x \leq 1\) \(0 \leq y \leq 2\).

Find the area of the region cut from the first quadrant by the cardioid \(r=1+\sin \theta.\)

Find. a. the mass of the solid. b. the center of mass. c. the moments of inertia about the coordinate axes. A solid cube in the first octant is bounded by the coordinate planes and by the planes \(x=1, y=1,\) and \(z=1 .\) The density of the cube is \(\delta(x, y, z)=x+y+z+1\).

Each gives an integral over a region in a Cartesian coordinate plane. Sketch the region and evaluate the integral. $$\int_{-\pi / 3}^{\pi / 3} \int_{0}^{\sec t} 3 \cos t d u d t \quad(\text { the } t u \text { -plane })$$

Find a value of the constant \(k\) so that \(\int_{1}^{2} \int_{0}^{3} k x^{2} y d x d y=1\).

Find the area of the region common to the interiors of the cardioids \(r=1+\cos \theta\) and \(r=1-\cos \theta.\)

Let \(D\) be the region bounded below by the cone \(z=\sqrt{x^{2}+y^{2}}\) and above by the plane \(z=1 .\) Set up the triple integrals in spherical coordinates that give the volume of \(D\) using the following orders of integration. a. \(d \rho d \phi d \theta\) b. \(d \phi d \rho d \theta\)

Each gives an integral over a region in a Cartesian coordinate plane. Sketch the region and evaluate the integral. $$\int_{0}^{3 / 2} \int_{1}^{4-2 u} \frac{4-2 u}{v^{2}} d v d u \quad \text { (the } u v \text { -plane) }$$

Evaluate \(\int_{-1}^{1} \int_{0}^{\pi / 2} x \sin \sqrt{y} d y d x\).

In polar coordinates, the average value of a function over a region \(R\) (Section 14.3 ) is given by $$\frac{1}{\operatorname{Area}(R)} \iint_{R} f(r, \theta) r d r d \theta$$ Find the average height of the hemispherical surface \(z=\sqrt{a^{2}-x^{2}-y^{2}}\) above the disk \(x^{2}+y^{2} \leq a^{2}\) in the \(x y\)-plane.

Find the mass of the solid bounded by the planes \(x+z=1\) \(x-z=-1, y=0,\) and the surface \(y=\sqrt{z} .\) The density of the solid is \(\delta(x, y, z)=2 y+5\).

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{1} \int_{2}^{4-2 x} d y d x$$

Use Fubini's Theorem to evaluate $$\int_{0}^{2} \int_{0}^{1} \frac{x}{1+x y} d x d y$$

The finite region bounded by the planes \(z=x, x+z=8, z=y\) \(y=8,\) and \(z=0\).

In polar coordinates, the average value of a function over a region \(R\) (Section 14.3 ) is given by $$\frac{1}{\operatorname{Area}(R)} \iint_{R} f(r, \theta) r d r d \theta$$ Find the average height of the (single) cone \(z=\sqrt{x^{2}+y^{2}}\) above the disk \(x^{2}+y^{2} \leq a^{2}\) in the \(x y\)-plane.

Find the mass of the solid region bounded by the parabolic surfaces \(z=16-2 x^{2}-2 y^{2}\) and \(z=2 x^{2}+2 y^{2}\) if the density of the solid is \(\delta(x, y, z)=\sqrt{x^{2}+y^{2}}\).

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{2} \int_{y-2}^{0} d x d y$$

The region cut from the solid elliptical cylinder \(x^{2}+4 y^{2} \leq 4\) by the \(x y\) -plane and the plane \(z=x+2\).

Use a software application to compute the integrals a. \(\int_{0}^{1} \int_{0}^{2} \frac{y-x}{(x+y)^{3}} d x d y\) b. \(\int_{0}^{2} \int_{0}^{1} \frac{y-x}{(x+y)^{3}} d y d x\) Explain why your results do not contradict Fubini's Theorem.

The region bounded in back by the plane \(x=0,\) on the front and sides by the parabolic cylinder \(x=1-y^{2},\) on the top by the paraboloid \(z=x^{2}+y^{2},\) and on the bottom by the \(x y\) -plane.

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{1} \int_{1-x}^{1-x^{2}} d y d x$$

If \(f(x, y)\) is continuous over \(R: a \leq x \leq b, c \leq y \leq d\) and $$F(x, y)=\int_{a}^{x} \int_{c}^{y} f(u, v) d v d u$$ on the interior of \(R,\) find the second partial derivatives \(F_{x y}\) and \(F_{y x}\)

In Exercises \(37-40,\) find the average value of \(F(x, y, z)\) over the given region. \(F(x, y, z)=x^{2}+9\) over the cube in the first octant bounded by the coordinate planes and the planes \(x=2, y=2,\) and \(z=2\)

Converting to a polar integral Integrate \( f(x, y)=\) \(\left[\ln \left(x^{2}+y^{2}\right)\right] / \sqrt{x^{2}+y^{2}}\) over the region \(1 \leq x^{2}+y^{2} \leq e.\)