Evaluate each integral.

$$

\int_{0}^{5}\left(x^{4} y+y\right) d x

$$

Donald Y.

Numerade Educator

Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$

$$

\int_{1}^{2} \int_{0}^{5}\left(x^{4} y+y\right) d x d y

$$

Donald Y.

Numerade Educator

Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$

$$

\int_{0}^{3} \int_{1}^{2}\left(x y^{3}-x\right) d y d x

$$

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Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$

$$

\int_{0}^{1} \int_{3}^{6} x \sqrt{x^{2}+3 y} d x d y

$$

Donald Y.

Numerade Educator

Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$

$$

\int_{0}^{3} \int_{4}^{5} x \sqrt{x^{2}+3 y} d y d x

$$

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Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$

$$

\int_{1}^{2} \int_{4}^{9} \frac{3+5 y}{\sqrt{x}} d x d y

$$

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Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$

$$

\int_{16}^{25} \int_{2}^{7} \frac{3+5 y}{\sqrt{x}} d y d x

$$

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Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$

$$

\int_{1}^{3} \int_{1}^{3} \frac{1}{x y} d y d x

$$

Donald Y.

Numerade Educator

Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$

$$

\int_{1}^{5} \int_{2}^{4} \frac{1}{y} d x d y

$$

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Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$

$$

\int_{2}^{4} \int_{3}^{5}\left(\frac{x}{y}+\frac{y}{3}\right) d x d y

$$

Donald Y.

Numerade Educator

Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$

$$

\int_{3}^{4} \int_{1}^{2}\left(\frac{6 x}{5}+\frac{y}{x}\right) d x d y

$$

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Find each double integral over the rectangular region $R$ with the given boundaries.

$$

\iint_{R}\left(3 x^{2}+4 y\right) d x d y ; \quad 0 \leq x \leq 3,1 \leq y \leq 4

$$

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Find each double integral over the rectangular region $R$ with the given boundaries.

$$

\iint_{R}\left(x^{2}+4 y^{3}\right) d y d x ; \quad 1 \leq x \leq 2,0 \leq y \leq 3

$$

Donald Y.

Numerade Educator

Find each double integral over the rectangular region $R$ with the given boundaries.

$$

\iint_{R} \sqrt{x+y} d y d x ; \quad 1 \leq x \leq 3,0 \leq y \leq 1

$$

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Find each double integral over the rectangular region $R$ with the given boundaries.

$$

\iint_{R} x^{2} \sqrt{x^{3}+2 y} d x d y ; \quad 0 \leq x \leq 2,0 \leq y \leq 3

$$

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Find each double integral over the rectangular region $R$ with the given boundaries.

$$

\iint_{R} \frac{3}{(x+y)^{2}} d y d x ; \quad 2 \leq x \leq 4,1 \leq y \leq 6

$$

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Find each double integral over the rectangular region $R$ with the given boundaries.

$$

\iint_{R} \frac{y}{\sqrt{2 x+5 y^{2}}} d x d y ; \quad 0 \leq x \leq 2,1 \leq y \leq 3

$$

Donald Y.

Numerade Educator

Find each double integral over the rectangular region $R$ with the given boundaries.

$$

\iint_{R} y e^{x+y^{2}} d x d y ; \quad 2 \leq x \leq 3,0 \leq y \leq 2

$$

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Find each double integral over the rectangular region $R$ with the given boundaries.

$$

\iint_{R} x^{2} e^{x^{3}+2 y} d x d y ; \quad 1 \leq x \leq 2,1 \leq y \leq 3

$$

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Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.

$$

z=8 x+4 y+10 ;-1 \leq x \leq 1,0 \leq y \leq 3

$$

Donald Y.

Numerade Educator

Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.

$$

z=3 x+10 y+20 ; \quad 0 \leq x \leq 3,-2 \leq y \leq 1

$$

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Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.

$$

z=x^{2} ; \quad 0 \leq x \leq 2,0 \leq y \leq 5

$$

Donald Y.

Numerade Educator

Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.

$$

z=\sqrt{y} ; 0 \leq x \leq 4,0 \leq y \leq 9

$$

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Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.

$$

z=x \sqrt{x^{2}+y} ; \quad 0 \leq x \leq 1,0 \leq y \leq 1

$$

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Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.

$$

z=y x \sqrt{x^{2}+y^{2}} ; \quad 0 \leq x \leq 4,0 \leq y \leq 1

$$

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Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.

$$

z=\frac{x y}{\left(x^{2}+y^{2}\right)^{2}} ; \quad 1 \leq x \leq 2,1 \leq y \leq 4

$$

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Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.

$$

z=e^{x+y} ; \quad 0 \leq x \leq 1,0 \leq y \leq 1

$$

Donald Y.

Numerade Educator

Although it is often true that a double integral can be evaluated by using either $d x$ or $d y$ first, sometimes one choice over the other makes the work easier. Evaluate the double integrals in Exercises 37 and 38 in the easiest way possible.

$$

\iint_{R} x e^{x y} d x d y ; \quad 0 \leq x \leq 2,0 \leq y \leq 1

$$

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Although it is often true that a double integral can be evaluated by using either $d x$ or $d y$ first, sometimes one choice over the other makes the work easier. Evaluate the double integrals in Exercises 37 and 38 in the easiest way possible.

$$

\iint_{R} 2 x^{3} e^{x^{2} y} d x d y ; \quad 0 \leq x \leq 1,0 \leq y \leq 1

$$

Donald Y.

Numerade Educator

Evaluate each double integral.

$$

\int_{2}^{4} \int_{2}^{x^{2}}\left(x^{2}+y^{2}\right) d y d x

$$

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Evaluate each double integral.

$$

\int_{0}^{2} \int_{0}^{3 y}\left(x^{2}+y\right) d x d y

$$

Donald Y.

Numerade Educator

Evaluate each double integral.

$$

\int_{0}^{4} \int_{0}^{x} \sqrt{x y} d y d x

$$

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Evaluate each double integral.

$$

\int_{1}^{4} \int_{0}^{x} \sqrt{x+y} d y d x

$$

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Evaluate each double integral.

$$

\int_{2}^{6} \int_{2 y}^{4 y} \frac{1}{x} d x d y

$$

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Evaluate each double integral.

$$

\int_{1}^{4} \int_{x}^{x^{2}} \frac{1}{y} d y d x

$$

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Evaluate each double integral.

$$

\int_{0}^{4} \int_{1}^{e^{x}} \frac{x}{y} d y d x

$$

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Evaluate each double integral.

$$

\int_{0}^{1} \int_{2 x}^{4 x} e^{x+y} d y d x

$$

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Use the region $R$ with the indicated boundaries to evaluate each double integral.

$$

\iint_{R}(5 x+8 y) d y d x ; \quad 1 \leq x \leq 3,0 \leq y \leq x-1

$$

Donald Y.

Numerade Educator

Use the region $R$ with the indicated boundaries to evaluate each double integral.

$$

\iint_{R}(2 x+6 y) d y d x ; \quad 2 \leq x \leq 4,2 \leq y \leq 3 x

$$

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Use the region $R$ with the indicated boundaries to evaluate each double integral.

$$

\iint_{R}\left(4-4 x^{2}\right) d y d x ; \quad 0 \leq x \leq 1,0 \leq y \leq 2-2 x

$$

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Use the region $R$ with the indicated boundaries to evaluate each double integral.

$$

\iint_{R} \frac{1}{x} d y d x ; \quad 1 \leq x \leq 2,0 \leq y \leq x-1

$$

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Use the region $R$ with the indicated boundaries to evaluate each double integral.

$$

\iint_{R} e^{x / y^{2}} d x d y ; \quad 1 \leq y \leq 2,0 \leq x \leq y^{2}

$$

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Use the region $R$ with the indicated boundaries to evaluate each double integral.

$$

\iint_{R}\left(x^{2}-y\right) d y d x ; \quad-1 \leq x \leq 1,-x^{2} \leq y \leq x^{2}

$$

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Use the region $R$ with the indicated boundaries to evaluate each double integral.

$$

\iint_{R} x^{3} y d y d x ; \quad R \text { bounded by } y=x^{2}, y=2 x

$$

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Use the region $R$ with the indicated boundaries to evaluate each double integral.

$$

\iint_{R} x^{2} y^{2} d x d y ; \quad R \text { bounded by } y=x, y=2 x, x=1

$$

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Use the region $R$ with the indicated boundaries to evaluate each double integral.

$$

\iint_{R} \frac{1}{y} d y d x ; \quad R \text { bounded by } y=x, y=\frac{1}{x}, x=2

$$

Donald Y.

Numerade Educator

Use the region $R$ with the indicated boundaries to evaluate each double integral.

$$

\iint_{R} e^{2 y / x} d y d x ; \quad R \text { bounded by } y=x^{2}, y=0, x=2

$$

Donald Y.

Numerade Educator

Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration, as in Exercises 37 and $38 .$

$$

\int_{0}^{\ln 2} \int_{e^{y}}^{2} \frac{1}{\ln x} d x d y

$$

Donald Y.

Numerade Educator

Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration, as in Exercises 37 and $38 .$

$$

\int_{0}^{2} \int_{y / 2}^{1} e^{x^{2}} d x d y

$$

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Recall from the Volume and Average Value section in the previous chapter that volume could be found with a single integral. In this section volume is found using a double integral. Explain when volume can be found with a single integral and when a double integral is needed.

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Give an example of a region that cannot be expressed by either of the forms shown in Figure $34 .$ (One example is the disk with a hole in the middle between the graphs of $x^{2}+y^{2}=1$ and $x^{2}+y^{2}=2$ in Figure $10 .$ )

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The idea of the average value of a function, discussed earlier for functions of the form $y=f(x),$ can be extended to functions of more than one independent variable. For a function $z=f(x, y),$ the average value of $f$ over a region $R$ is defined as

$$\frac{1}{A} \iint_{R} f(x, y) d x d y$$

where $A$ is the area of the region $R .$ Find the average value for each function over the regions $R$ having the given boundaries.

$$

f(x, y)=6 x y+2 x ; \quad 2 \leq x \leq 5,1 \leq y \leq 3

$$

Donald Y.

Numerade Educator

The idea of the average value of a function, discussed earlier for functions of the form $y=f(x),$ can be extended to functions of more than one independent variable. For a function $z=f(x, y),$ the average value of $f$ over a region $R$ is defined as

$$\frac{1}{A} \iint_{R} f(x, y) d x d y$$

where $A$ is the area of the region $R .$ Find the average value for each function over the regions $R$ having the given boundaries.

$$

f(x, y)=x^{2}+y^{2} ; \quad 0 \leq x \leq 2,0 \leq y \leq 3

$$

Donald Y.

Numerade Educator

The idea of the average value of a function, discussed earlier for functions of the form $y=f(x),$ can be extended to functions of more than one independent variable. For a function $z=f(x, y),$ the average value of $f$ over a region $R$ is defined as

$$\frac{1}{A} \iint_{R} f(x, y) d x d y$$

where $A$ is the area of the region $R .$ Find the average value for each function over the regions $R$ having the given boundaries.

$$

f(x, y)=e^{-5 y+3 x} ; \quad 0 \leq x \leq 2,0 \leq y \leq 2

$$

Donald Y.

Numerade Educator

The idea of the average value of a function, discussed earlier for functions of the form $y=f(x),$ can be extended to functions of more than one independent variable. For a function $z=f(x, y),$ the average value of $f$ over a region $R$ is defined as

$$\frac{1}{A} \iint_{R} f(x, y) d x d y$$

where $A$ is the area of the region $R .$ Find the average value for each function over the regions $R$ having the given boundaries.

$$

f(x, y)=e^{2 x+y} ; \quad 1 \leq x \leq 2,2 \leq y \leq 3

$$

Donald Y.

Numerade Educator

Packaging The manufacturer of a fruit juice drink has decided to try innovative packaging in order to revitalize sagging sales. The fruit juice drink is to be packaged in containers in the shape of tetrahedra in which three edges are perpendicular, as shown in the figure on the next page. Two of the perpendicular edges will be 3 in. long, and the third edge will be 6 in. long. Find the volume of the container. (Hint: The equation of the plane shown in the figure is $z=f(x, y)=6-2 x-2 y . )$

Donald Y.

Numerade Educator

Average Cost A company's total cost for operating its two warehouses is

$$C(x, y)=\frac{1}{9} x^{2}+2 x+y^{2}+5 y+100$$

dollars, where $x$ represents the number of units stored at the first warehouse and $y$ represents the number of units stored at the second. Find the average cost to store a unit if the first warehouse has between 40 and 80 units, and the second has between 30 and 70 units. (Hint: Refer to Exercises $61-64 . )$

Sajin S.

Numerade Educator

Average Production A production function is given by

$$P(x, y)=500 x^{0.2} y^{0.8}$$

where $x$ is the number of units of labor and $y$ is the number of units of capital. Find the average production level if $x$ varies from 10 to 50 and $y$ from 20 to $40 .$ (Hint: Refer to Exercises $61-64 . )$

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Average Profit The profit (in dollars) from selling $x$ units of one product and $y$ units of a second product is

$$P=-(x-100)^{2}-(y-50)^{2}+2000$$

The weekly sales for the first product vary from 100 units to 150 units, and the weekly sales for the second product vary from 40 units to 80 units. Estimate average weekly profit for these two products. (Hint: Refer to Exercises $61-64 .$ )

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Average Revenue A company sells two products. The demand functions of the products are given by

$$q_{1}=300-2 p_{1} \quad \text { and } \quad q_{2}=500-1.2 p_{2}$$

where $q_{1}$ units of the first product are demanded at price $p_{1}$ and $q_{2}$ units of the second product are demanded at price $p_{2} .$ The total revenue will be given by

$$R=q_{1} p_{1}+q_{2} p_{2}$$

Find the average revenue if the price $p_{1}$ varies from $\$ 25$ to $\$ 50$ and the price $p_{2}$ varies from $\$ 50$ to $\$ 75 .$ (Hint: Refer to Exercises $61-64 .$ )

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Time In an exercise earlier in this chapter, we saw that the time (in hours) that a branch of Amalgamated Entities needs to spend to meet the quota set by the main office can be approximated by

$$T(x, y)=x^{4}+16 y^{4}-32 x y+40$$

where $x$ represents how many thousands of dollars the factory spends on quality control and $y$ represents how many thousands of dollars they spend on consulting. Find the average time if the amount spent on quality control varies from $\$ 0$ to $\$ 4000$ and the amount spent on consulting varies from $\$ 0$ to $\$ 2000$ . (Hint: Refer to Exercises $61-64 . )$

Sajin S.

Numerade Educator

Profit In an exercise earlier in this chapter, we saw that the profit (in thousands of dollars) that Aunt Mildred's Metalworks earns from producing $x$ tons of steel and $y$ tons of aluminum can be approximated by

$$P(x, y)=36 x y-x^{3}-8 y^{3}$$

Find the average profit if the amount of steel produced varies from 0 to 8 tons, and the amount of aluminum produced varies from 0 to 4 tons. (Hint: Refer to Exercises $61-64 .$ )

Sajin S.

Numerade Educator