Double Integrals over Rectangles

Calculus 3

131 Practice Problems

20:31
Calculus with Applications

Profit In Exercise 38 of Section $9.3,$ we saw that the profit (in thousands of dollars) that Aunt Mildred's Meta lworks 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 $61-64 . )$

Multivariable Calculus
Double Integrals
03:34
Calculus with Applications

Time In Exercise 39 of Section $9.3,$ 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 pends on quality control and $y$ represents how many thousands of dollars they spend on consulting. Find the average time if the
mount spent on quality control varies from $\$ 0$to$\$4000$ andthe amount spent on consulting varies from $\$ 0$to$\$2000 .$ Hint: Refer to Exercises $61-64$ .

Multivariable Calculus
Double Integrals
33:45
Calculus with Applications

Average Revenue A company sells two products. The demand functions of the products are given by
$q_{1}=300-2 p_{1} \quad$ 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 $61-64 . )$

Multivariable Calculus
Double Integrals
05:57
Calculus with Applications

Average Profit one product and $v$ 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 $61-64$ )

Multivariable Calculus
Double Integrals
18:13
Calculus with Applications

Average Production $\mathrm{A}$ company's production function is
given by
$P(x, y)=200 x^{0.25} y^{0.75}$
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 20 and $y$ from 50 to 100 .

Multivariable Calculus
Double Integrals
07:06
Calculus with Applications

Average cost A restaurant's cost function is approximated by
$C(x, y)=\frac{1}{4} x^{2}+4 x+2 y^{2}+y+10$
dollars, where $x$ represents the cost of labor per hour and $y$ represents the average cost of materials per dish. Find the average cost of the restaurant per hour if the cost of labor per hour in the
restaurant is between $\$ 30$and$\$60,$ and the cost of material perdish is between $\$ 50$and$\$80 .$

Multivariable Calculus
Double Integrals
02:20
Calculus with Applications

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

Multivariable Calculus
Double Integrals
17:18
Calculus with Applications

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

Multivariable Calculus
Double Integrals
02:27
Calculus with Applications

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

Multivariable Calculus
Double Integrals
16:48
Calculus with Applications

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

Multivariable Calculus
Double Integrals
03:05
Calculus with Applications

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 $10 . )$
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 foreach function over the regions $R$ having the given boundaries.

Multivariable Calculus
Double Integrals
03:27
Calculus with Applications

Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration, as in Exercises 37 and $38 .$
$$\int_{0}^{2} \int_{y / 2}^{1} e^{x^{2}} d x d y$$

Multivariable Calculus
Double Integrals
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