Chapter 13

Set up the iterated integral that computes the surface area of the given surface over the region \(R .\) \(f(x, y)=\frac{1}{e^{x^{2}+1}} ; \quad R\) is the rectangle bounded by \(-5 \leq x \leq 5\) and \(0 \leq y \leq 1\).

In Exercises \(3-10\), a function \(f(x, y)\) is given and a region \(R\) of the \(x-y\) plane is described. Set up and evaluate \(\iint_{R} f(x, y) d A\) using polar coordinates. \(f(x, y)=(x-y) /(x+y) ; R\) is the region enclosed by the lines \(y=x, y=0\) and the circle \(x^{2}+y^{2}=1\) in the first quadrant.

Point masses are given along a line or in the plane. Find the center of \(\operatorname{mass} \bar{x}\) or \((\bar{x}, \bar{y}),\) as appropriate. (All masses are in grams and distances are in cm.) $$ \begin{array}{l} m_{1}=1 \text { at }(-1,-1) ; \quad m_{2}=2 \text { at }(-1,1) ; \\ m_{3}=2 \text { at }(1,1) ; \quad m_{4}=1 \text { at }(1,-1) \end{array} $$

Evaluate the integral and subsequent iterated integral. (a) \(\int_{0}^{x}\left(\frac{1}{1+x^{2}}\right) d y\) (b) \(\int_{1}^{2} \int_{0}^{x}\left(\frac{1}{1+x^{2}}\right) d y d x\)

(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration. $$ \int_{0}^{9} \int_{y / 3}^{\sqrt{y}}\left(x y^{2}\right) d x d y $$

A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{\pi / 2} \int_{0}^{2} \int_{0}^{2} r d z d r d \theta $$

Find the area of the given surface over the region \(R\). \(f(x, y)=3 x-7 y+2 ; R\) is the rectangle with opposite corners (-1,0) and (1,3).

In Exercises \(11-14,\) an iterated integral in rectangular coordinates is given. Rewrite the integral using polar coordinates and evaluate the new double integral. $$ \int_{0}^{5} \int_{-\sqrt{25-x^{2}}}^{\sqrt{25-x^{2}}} \sqrt{x^{2}+y^{2}} d y d x $$

In Exercises \(11-18,\) find the mass/weight of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\). \(R\) is the rectangle with corners (1,-3),(1,2),(7,2) and (7,-3)\(; \delta(x, y)=5 \mathrm{gm} / \mathrm{cm}^{2}\)

(a) Sketch the region \(R\) given by the problem. (b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region \(R\) (c) Evaluate one of the iterated integrals to find the signed volume under the surface \(z=f(x, y)\) over the region \(R .\) \(\iint_{R} x^{2} y d A,\) where \(R\) is bounded by \(y=\sqrt{x}\) and \(y=x^{2} .\)

A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{3}^{4} \int_{0}^{5} r d z d r d \theta $$

Find the area of the given surface over the region \(R\). \(f(x, y)=2 x+2 y+2 ; R\) is the triangle with corners (0,0) (1,0) and (0,1)

In Exercises \(11-14,\) an iterated integral in rectangular coordinates is given. Rewrite the integral using polar coordinates and evaluate the new double integral. $$ \int_{-4}^{4} \int_{-\sqrt{16-y^{2}}}^{0}(2 y-x) d x d y $$

Find the mass/weight of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\). \(R\) is the rectangle with corners (1,-3),(1,2),(7,2) and (7,-3)\(; \delta(x, y)=\left(x+y^{2}\right) \mathrm{gm} / \mathrm{cm}^{2}\)

(a) Sketch the region \(R\) given by the problem. (b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region \(R\) (c) Evaluate one of the iterated integrals to find the signed volume under the surface \(z=f(x, y)\) over the region \(R .\) \(\iint_{R} x^{2} y d A,\) where \(R\) is bounded by \(y=\sqrt[3]{x}\) and \(y=x^{3} .\)

A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{0}^{1} \int_{0}^{1-r} r d z d r d \theta $$

Find the area of the given surface over the region \(R\). \(f(x, y)=x^{2}+y^{2}+10 ; R\) is bounded by the circle \(x^{2}+y^{2}=\) \(16 .\)

In Exercises \(11-14,\) an iterated integral in rectangular coordinates is given. Rewrite the integral using polar coordinates and evaluate the new double integral. $$ \int_{0}^{2} \int_{y}^{\sqrt{8-y^{2}}}(x+y) d x d y $$

Find the mass/weight of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\). \(R\) is the triangle with corners \((-1,0),(1,0),\) and (0,1)\(;\) \(\delta(x, y)=2 \mathrm{lb} / \mathrm{in}^{2}\)

(a) Sketch the region \(R\) given by the problem. (b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region \(R\) (c) Evaluate one of the iterated integrals to find the signed volume under the surface \(z=f(x, y)\) over the region \(R .\) \(\iint_{R} x^{2}-y^{2} d A,\) where \(R\) is the rectangle with corners (-1,-1),(1,-1),(1,1) and (-1,1)

A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{\pi} \int_{0}^{1} \int_{0}^{2-r} r d z d r d \theta $$

Find the area of the given surface over the region \(R\). \(f(x, y)=-2 x+4 y^{2}+7\) over \(R,\) the triangle bounded by \(y=-x, y=x, 0 \leq y \leq 1\).

In Exercises \(11-14,\) an iterated integral in rectangular coordinates is given. Rewrite the integral using polar coordinates and evaluate the new double integral. $$ \begin{array}{l} \int_{-2}^{-1} \int_{0}^{\sqrt{4-x^{2}}}(x+5) d y d x+\int_{-1}^{1} \int_{\sqrt{1-x^{2}}}^{\sqrt{4-x^{2}}}(x+5) d y d x+ \\ \int_{1}^{2} \int_{0}^{\sqrt{4-x^{2}}}(x+5) d y d x \end{array} $$

Find the mass/weight of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\). \(R\) is the triangle with corners \((0,0),(1,0),\) and (0,1)\(;\) \(\delta(x, y)=\left(x^{2}+y^{2}+1\right) \mathrm{Ib} / \mathrm{in}^{2}\)

(a) Sketch the region \(R\) given by the problem. (b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region \(R\) (c) Evaluate one of the iterated integrals to find the signed volume under the surface \(z=f(x, y)\) over the region \(R .\) \(\iint_{R} y e^{x} d A,\) where \(R\) is bounded by \(x=0, x=y^{2}\) and \(y=1 .\)

A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{\pi} \int_{0}^{3} \int_{0}^{\sqrt{9-r^{2}}} r d z d r d \theta $$

In Exercises \(15-16,\) special double integrals are presented that are especially well suited for evaluation in polar coordinates. Consider \(\iint_{f} e^{-\left(x^{2}+y^{2}\right)} d A .\) (a) Why is this integral difficult to evaluate in rectangular coordinates, regardless of the region \(R ?\) (b) Let \(R\) be the region bounded by the circle of radius \(a\) centered at the origin. Evaluate the double integral using polar coordinates. (c) Take the limit of your answer from \((b),\) as \(a \rightarrow \infty\). What does this imply about the volume under the surface of \(e^{-\left(x^{2}+y^{2}\right)}\) over the entire \(x-y\) plane?

Find the area of the given surface over the region \(R\). \(f(x, y)=x^{2}+y\) over \(R,\) the triangle bounded by \(y=2 x\) \(y=0\) and \(x=2\).

Find the mass/weight of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\). \(R\) is the disk centered at the origin with radius 2; \(\delta(x, y)=\) \((x+y+4) \mathrm{kg} / \mathrm{m}^{2}\)

(a) Sketch the region \(R\) given by the problem. (b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region \(R\) (c) Evaluate one of the iterated integrals to find the signed volume under the surface \(z=f(x, y)\) over the region \(R .\) \(\iint_{R}(6-3 x-2 y) d A,\) where \(R\) is bounded by \(x=0, y=0\) and \(3 x+2 y=6\)

A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-r^{2}}+b} r d z d r d \theta $$

Set up the triple integrals that give the volume of \(D\) in all 6 orders of integration, and find the volume of \(D\) by evaluating the indicated triple integral. \(D\) is bounded by the coordinate planes and by \(z=1-y / 3\) and \(z=1-x\) Evaluate the triple integral with order \(d x d y d z\).

In Exercises \(15-16,\) special double integrals are presented that are especially well suited for evaluation in polar coordinates. The surface of a right circular cone with height \(h\) and base radius \(a\) can be described by the equation \(f(x, y)=\) \(h-h \sqrt{\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}},\) where the tip of the cone lies at \((0,0, h)\) and the circular base lies in the \(x\) -y plane, centered at the origin. Confirm that the volume of a right circular cone with height \(h\) and base radius \(a\) is \(V=\frac{1}{3} \pi a^{2} h\) by evaluating \(\iint_{R} f(x, y) d A\) in polar coordinates.

Find the area of the given surface over the region \(R\). \(f(x, y)=\frac{2}{3} x^{3 / 2}+2 y^{3 / 2}\) over \(R,\) the rectangle with opposite corners (0,0) and (1,1).

Find the mass/weight of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\). \(R\) is the circle sector bounded by \(x^{2}+y^{2}=25\) in the first quadrant; \(\delta(x, y)=\left(\sqrt{x^{2}+y^{2}}+1\right) \mathrm{kg} / \mathrm{m}^{2}\)

(a) Sketch the region \(R\) given by the problem. (b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region \(R\) (c) Evaluate one of the iterated integrals to find the signed volume under the surface \(z=f(x, y)\) over the region \(R .\) \(\iint_{R} e^{y} d A,\) where \(R\) is bounded by \(y=\ln x\) and \(y=\frac{1}{e-1}(x-1)\).

A triple integral in spherical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{\pi / 2} \int_{0}^{\pi} \int_{0}^{1} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi $$

Evaluate the triple integral. $$ \int_{-\pi / 2}^{\pi / 2} \int_{0}^{\pi} \int_{0}^{\pi}(\cos x \sin y \sin z) d z d y d x $$

Find the area of the given surface over the region \(R\). \(f(x, y)=10-2 \sqrt{x^{2}+y^{2}}\) over \(R,\) bounded by the circle \(x^{2}+y^{2}=25\). (This is the cone with height 10 and base radius 5 ; be sure to compare your result with the known formula.)

Iterated integrals are given that compute the area of a region \(R\) in the \(x\) -y plane. Sketch the region \(R\), and give the iterated integral(s) that give the area of \(R\) with the opposite order of integration. $$ \int_{-2}^{2} \int_{0}^{4-x^{2}} d y d x $$

(a) Sketch the region \(R\) given by the problem. (b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region \(R\) (c) Evaluate one of the iterated integrals to find the signed volume under the surface \(z=f(x, y)\) over the region \(R .\) \(\iint_{R}\left(x^{3} y-x\right) d A,\) where \(R\) is the half of the circle \(x^{2}+y^{2}=9\) in the first and second quadrants.

A triple integral in spherical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{\pi} \int_{0}^{\pi} \int_{1}^{1.1} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi $$

Evaluate the triple integral. $$ \int_{0}^{1} \int_{0}^{x} \int_{0}^{x+y}(x+y+z) d z d y d x $$

Find the area of the given surface over the region \(R\). Find the surface area of the sphere with radius 5 by doubling the surface area of \(f(x, y)=\sqrt{25-x^{2}-y^{2}}\) over \(R\), bounded by the circle \(x^{2}+y^{2}=25 .\) (Be sure to compare your result with the known formula.)

Find the mass/weight of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\). \(R\) is the annulus in the first and second quadrants bounded by \(x^{2}+y^{2}=9\) and \(x^{2}+y^{2}=36 ; \delta(x, y)=\sqrt{x^{2}+y^{2}} \mid b / f t^{2}\)

(a) Sketch the region \(R\) given by the problem. (b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region \(R\) (c) Evaluate one of the iterated integrals to find the signed volume under the surface \(z=f(x, y)\) over the region \(R .\) \(\iint_{R}(4-3 y) d A,\) where \(R\) is bounded by \(y=0, y=x / e\) and \(y=\ln x\).

Iterated integrals are given that compute the area of a region \(R\) in the \(x\) -y plane. Sketch the region \(R\), and give the iterated integral(s) that give the area of \(R\) with the opposite order of integration. $$ \int_{0}^{1} \int_{5-5 x}^{5-5 x^{2}} d y d x $$

A triple integral in spherical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{2} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi $$

Evaluate the triple integral. $$ \int_{0}^{\pi} \int_{0}^{1} \int_{0}^{z}(\sin (y z)) d x d y d z $$

In Exercises \(19-26,\) find the center of mass of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\) Note: these are the same lamina as in Exercises \(11-18\). \(R\) is the rectangle with corners (1,-3),(1,2),(7,2) and (7,-3)\(; \delta(x, y)=5 \mathrm{gm} / \mathrm{cm}^{2}\)