Chapter 7

Find the volume of the solid under the surface \(z=f(x, y)\) and over the given region \(R\).$$ f(x, y)=x e^{-y} ; $$ \(R: 0 \leq x \leq 1,0 \leq y \leq 2\)

Find the volume of the solid under the surface \(z=f(x, y)\) and over the given region \(R\).\(f(x, y)=(1-x)(4-y)\) \(R: 0 \leq x \leq 1,0 \leq y \leq 4\)

Find the volume of the solid under the surface \(z=f(x, y)\) and over the given region \(R\).\(f(x, y)=2 x+y ; R\) is bounded by \(y=x, y=2-x\), and \(y=0\).

Find the volume of the solid under the surface \(z=f(x, y)\) and over the given region \(R\).\(f(x, y)=e^{y^{2}} ; R\) is bounded by \(x=2 y, x=0\), and \(y=1\).

Find the volume of the solid under the surface \(z=f(x, y)\) and over the given region \(R\).\(f(x, y)=x+1 ; R\) is bounded by \(y=8-x^{2}\) and \(y=x^{2}\).

Find the volume of the solid under the surface \(z=f(x, y)\) and over the given region \(R\).\(f(x, y)=4 x e^{y} ; R\) is bounded by \(y=2 x, y=2\), and \(x=0\).

Find the average value of the function \(f(x, y)\) over the given region \(R\).\(f(x, y)=x y(x-2 y)\) \(R:-2 \leq x \leq 3,-1 \leq y \leq 2\)

Find the average value of the function \(f(x, y)\) over the given region \(R\).$$ f(x, y)=\frac{y}{x}+\frac{x}{y} ; $$ \(R: 1 \leq x \leq 4,1 \leq y \leq 3\)

Find the average value of the function \(f(x, y)\) over the given region \(R\).$$ f(x, y)=x y e^{x^{2} y} $$ \(R: 0 \leq x \leq 1,0 \leq y \leq 2\)

Find the average value of the function \(f(x, y)\) over the given region \(R\).$$ f(x, y)=\frac{\ln x}{x y} $$ \(R: 1 \leq x \leq 2,2 \leq y \leq 3\)

Find the average value of the function \(f(x, y)\) over the given region \(R\).\(f(x, y)=6 x y ; R\) is the triangle with vertices \((0,0),(0,1),(3,1)\).

Find the average value of the function \(f(x, y)\) over the given region \(R\).\(f(x, y)=e^{x^{2}} ; R\) is the triangle with vertices \((0,0),(1,0),(1,1)\).

Find the average value of the function \(f(x, y)\) over the given region \(R\).\(f(x, y)=x ; R\) is the region bounded by \(y=4-x^{2}\) and \(y=0\).

Find the average value of the function \(f(x, y)\) over the given region \(R\).\(f(x, y)=e^{x} y^{-1 / 2} ; R\) is the region bounded by \(x=\sqrt{y}, y=0\), and \(x=1\).

Evaluate the double integral over the specified region \(R\). Choose the order of integration carefully.\(\iint_{R} \frac{\ln (x y)}{y} d A ; R: 1 \leq x \leq 3,2 \leq y \leq 5\)

Evaluate the double integral over the specified region \(R\). Choose the order of integration carefully.\(\iint_{R} y e^{x y} d A ; R:-1 \leq x \leq 1,1 \leq y \leq 2\)

Evaluate the double integral over the specified region \(R\). Choose the order of integration carefully.\(\iint_{R} e^{x^{3}} d A ; R: \sqrt{y} \leq x \leq 1,0 \leq y \leq 1\)

PRODUCTION A bicycle dealer has found that if 10 -speed bicycles are sold for \(x\) dollars apiece and the price of gasoline is \(y\) cents per gallon, then approximately $$ Q(x, y)=200-24 \sqrt{x}+4(0.1 y+3)^{3 / 2} $$ bicycles will be sold each month. If the price of bicycles varies between \(\$ 289\) and \(\$ 324\) during a typical month, and the price of gasoline varies between \(\$ 2.96\) and \(\$ 3.05\), approximately how many bicycles will be sold each month on average?

AVERAGE PROFIT A manufacturer estimates that when \(x\) units of a particular commodity are sold domestically and \(y\) units are sold to foreign markets, the profit is given by $$ \begin{aligned} P(x, y)=&(x-30)(70+5 x-4 y) \\ &+(y-40)(80-6 x+7 y) \end{aligned} $$ hundred dollars. If monthly domestic sales vary between 100 and 125 units and foreign sales between 70 and 89 units, what is the average monthly profit?

PROPERTY VALUE A community is laid out as a rectangular grid in relation to two main streets that intersect at the city center. Each point in the community has coordinates \((x, y)\) in this grid, for \(-10 \leq x \leq 10,-8 \leq y \leq 8\) with \(x\) and \(y\) measured in miles. Suppose the value of the land located at the point \((x, y)\) is \(V\) thousand dollars, where $$ V(x, y)=(250+17 x) e^{-0.01 x-0.05 y} $$ Estimate the value of the block of land occupying the rectangular region \(1 \leq x \leq 3,0 \leq y \leq 2\).

AVERAGE ELEVATION A map of a small regional park is a rectangular grid, bounded by the lines \(x=0, x=4, y=0\), and \(y=3\), where units are in miles. It is found that the elevation above sea level at each point \((x, y)\) in the park is given by $$ E(x, y)=90\left(2 x+y^{2}\right) \text { feet } $$ Find the average elevation in the park. (Remember, \(1 \mathrm{mi}=5,280 \mathrm{feet}\) )

AVERAGE RESPONSE TO STIMULI In a psychological experiment, \(x\) units of stimulus A and \(y\) units of stimulus B are applied to a subject, whose performance on a certain task is then measured by the function $$ P(x, y)=10+x y e^{1-x^{2}-y^{2}} $$ Suppose \(x\) varies between 0 and 1 while \(y\) varies between 0 and 3 . What is the subject's average response to the stimuli?

CONSTRUCTION A storage bin is to be constructed in the shape of the solid bounded above by the surface $$ z=20-x^{2}-y^{2} $$ below by the \(x y\) plane, and on the sides by the plane \(y=0\) and the parabolic cylinder \(y=4-x^{2}\), where \(x, y\), and \(z\) are in meters. Find the volume of the bin.

ARCHITECTURAL DESIGN A building is to have a curved roof above a rectangular base. In relation to a rectangular grid, the base is the rectangular region \(-30 \leq x \leq 30,-20 \leq y \leq 20\), where \(x\) and \(y\) are measured in meters. The height of the roof above each point \((x, y)\) in the base is given by $$ h(x, y)=12-0.003 x^{2}-0.005 y^{2} $$ a. Find the volume of the building. b. Find the average height of the roof.

EXPOSURE TO DISEASE The likelihood that a person with a contagious disease will infect others in a social situation may be assumed to be a function \(f(s)\) of the distance \(s\) between individuals. Suppose contagious individuals are uniformly distributed throughout a rectangular region \(R\) in the \(x y\) plane. Then the likelihood of infection for someone at the origin \((0,0)\) is proportional to the exposure index \(E\), given by the double integral $$ E=\iint_{R} f(s) d A $$ where \(s=\sqrt{x^{2}+y^{2}}\) is the distance between \((0,0)\) and \((x, y)\). Find \(E\) for the case where $$ f(s)=1-\frac{s^{2}}{9} $$ and \(R\) is the square $$ R:-2 \leq x \leq 2,-2 \leq y \leq 2 $$

POPULATION The population density is \(f(x, y)=2,500 e^{-0.01 x-0.02 y}\) people per square mile at each point \((x, y)\) within the triangular region \(R\) with vertices \((-5,-2)\), \((0,3)\), and \((5,-2)\). Find the total population in the region \(R\).

POPULATION The population density is \(f(x, y)=1,000 y^{2} e^{-0.01 x}\) people per square mile at each point \((x, y)\) within the region \(R\) bounded by the parabola \(x=y^{2}\) and the vertical line \(x=4\). Find the total population in the region \(R\).

Find the area of the region bounded above by the curve (ellipse) \(4 x^{2}+3 y^{2}=7\) and below by the parabola \(y=x^{2}\).

Find the volume of the solid bounded above by the graph of \(f(x, y)=x^{2} e^{-x y}\) and below by the rectangular region \(R: 0 \leq x \leq 2,0 \leq y \leq 3\).

Find the average value of \(f(x, y)=x y \ln \left(\frac{y}{x}\right)\) over the rectangular region bounded by the lines \(x=1\), \(x=2, y=1\), and \(y=3\).