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A closed, bounded region \(R\) that has a boundary consisting of two smooth curves, \(y=g_{1}(x)\) and \(y=g_{2}(x),\) defined on \(a\leq x\leq b,\) where \(g_{1}(x)\leq g_{2}(x)\) on \([a,b] \) and possibly a portion of the vertical lines \(x=a\) and \(x=b,\) is called a(n) ________ region.

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True or False The double integral of a function that is continuous on a closed, bounded, \(y\)-simple region \(R\) can be expressed as an iterated integral.

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True or False The region \(R\) enclosed by the parabola \(y=x^{2}+1\) and the line \(y=3x+1\) is both \(x\)-simple and \(y\)-simple.

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Explain why \(\int_{0}^{1}\left[ \int_{0}^{x}f(x,y)\, {\it dy}\right]\! {\it dx}\neq \int_{0}^{1}\left[ \int_{0}^{y}f(x,y)\,{\it dx}\right]\! {\it dy}.\)

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\(\displaystyle\int_{0}^{1}\left[ \int_{x^{2}}^{\sqrt{x}}\,{\it dy}\right]\! {\it dx}\)

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\(\displaystyle\int_{0}^{1}\left[ \int_{y^{2}}^{\sqrt{y}}x\,{\it dx}\right]\! {\it dy}\)

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\(\displaystyle\int_{-1}^{2}\left[\int_{y^{2}}^{y+2}\,{\it dx}\right]\! {\it dy}\)

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\(\displaystyle\int_{0}^{1}\left[ \int_{x}^{2x}y\,{\it dy}\right]\! {\it dx}\)

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\(\displaystyle\int_{0}^{1}\left[ \int_{x}^{3x}xy\,{\it dy}\right]\! {\it dx}\)

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\(\displaystyle\int_{0}^{1}\left[ \int_{y}^{\sqrt{y}}x^{2}y\,{\it dx}\right]\! {\it dy}\)

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\(\int_{0}^{1}\left[ \int_{1}^{e^{y}}\dfrac{y}{x}\,{\it dx}\right]\! {\it dy}\)

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\(\int_{0}^{1}\left[\int_{0}^{x^{2}}xe^{y}\,{\it dy}\right]\! {\it dx}\)

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\(\int_{0}^{2}\left[ \int_{y}^{2y}xy\,{\it dx}\right]\! {\it dy}\)

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\(\int_{2}^{4}\left[ \int_{1}^{y^{2}}\dfrac{y}{x^{2}}\,{\it dx}\right]\! {\it dy}\)

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\(\int_{0}^{1}\left[ \int_{x^{2}}^{x}\sqrt{xy}\,{\it dy}\right]\! {\it dx}\)

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\(\int_{0}^{1}\left[\int_{x^{2}}^{x}\sqrt{x}\,{\it dy}\right]\! {\it dx}\)

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\(\int_{0}^{1}\left[ \int_{x}^{1}\dfrac{1}{x^{2}+1}\,{\it dy}\right]\! {\it dx}\)

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\(\int_{0}^{1}\left[ \int_{y}^{\sqrt{y}}(x^{2}+y^{2})\,{\it dx}\right]\! {\it dy}\)

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\(\int_{1}^{2}\left[ \int_{0}^{\ln x}xe^{y}\,{\it dy}\right]\! {\it dx}\)

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\(\int_{2}^{3}\left[ \int_{0}^{1/y}\ln y\,{\it dx}\right]\! {\it dy}\)

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\(\displaystyle\iint\limits_{\kern-3ptR}(x+y)\,{\it dA},\) where \(R\) is the region enclosed by \(y=x^{2}\) and \(y^{2}=8x\)

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\(\displaystyle\iint\limits_{\kern-3ptR}(x^{2}-y^{2})\,{\it dA},\) where \(R\) is the region enclosed by \(y=x\) and \(y=x^{2}\)

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\(\displaystyle\iint\limits_{\kern-3ptR}y^{2}\,{\it dA},\) where \(R\) is the region enclosed by \(y=2-x\) and \(y=x^{2}\)

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\(\displaystyle\iint\limits_{\kern-3ptR}xy\,{\it dA},\) where \(R\) is the region enclosed by \(y^{2}=x+1\) and \(y=1-x\)

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Enclosed by the graphs of \(y=x^{3}\) and \(y=x^{2}\)

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Enclosed by the graphs of \(y=2 \sqrt{x}\) and \(y=\dfrac{x^{2}}{4}\)

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Enclosed by the graphs of \(y=x^{2}-9\) and \(y=9-x^{2}\)

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Enclosed by the graphs of \(x^{2}+y^{2}=16\) and \(y^{2}=6x\)

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Enclosed by the graph \(y=\dfrac{1}{\sqrt{x-1}}\), the \(x\) -axis, and the lines \(x=2\) and \(x=5\)

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Enclosed by the graphs of \(y=x^{3/2}\) and \(y=x\)

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Enclosed by the line \(x+y=3\) and the hyperbola \(xy=2\)

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Enclosed by the hyperbola \(xy=\sqrt{3}\) and the circle \(x^{2}+y^{2}=4\), in the first quadrant only

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\(\int_{0}^{1}\left[ \int_{0}^{x}f(x,y)\,{\it dy}\right]\! {\it dx}\)

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\(\int_{0}^{2}\left[ \int_{y^{2}}^{2y}f(x,y)\,{\it dx}\right]\! {\it dy}\)

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\(\int_{0}^{a}\left[ \int_{0}^{\sqrt{a^{2}-y^{2}}}f(x,y)\,{\it dx}\right]\! {\it dy}\)

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\(\int_{0}^{2\sqrt[3]{2}}\left[ \int_{x^{2}/4}^{\sqrt{x}}f(x,y)\,{\it dy}\right]\! {\it dx}\)

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\(\int_{0}^{16}\left[\int^{y^{1/4}}_{y/8} f(x,y)\,{\it dx}\right]\! {\it dy}\)

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\(\int_{2}^{5}\left[ \int_{x^{2}-6x+9}^{x-1}f(x,y)\,{\it dy}\right]\! {\it dx}\)

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\(\int_{1}^{2}\left[ \int_{0}^{\ln y}f(x,y)\,{\it dx}\right]\! {\it dy}\)

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\(\int_{1}^{e}\left[ \int_{\ln x}^{1}f(x,y)\,{\it dy} \right]\! {\it dx}\)

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\(\displaystyle\iint\limits_{\kern-3ptR}[ f(x,y) -g(x,y) ] \, {\it dA}\)

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\(\displaystyle\iint\limits_{\kern-3ptR}4f(x,y) \, {\it dA}\)

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\(\displaystyle\iint\limits_{\kern-3ptR}[ 3f(x,y) +4g(x,y) ] \, {\it dA}\)

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\(\displaystyle\iint\limits_{\kern-3ptR} [ 2f(x,y) +5g(x,y) ] \, {\it dA}\)

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Area Find the area of the region \(R\) in the first quadrant enclosed by the graphs of \(y=6x-x^{2}\) and \(y=4x-8\) by partitioning \(R\) with a horizontal line through the point \((4,8)\). Refer to Figure 20.

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Area Use double integration to find the area in the first quadrant enclosed by the parabola \(x^{2}=9y\), the \(y\)-axis, and the circle \(x^{2}+y^{2}=10.\)

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The tetrahedron enclosed by the plane \(x+2y+z=2\) and the coordinate planes

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Enclosed by the elliptic paraboloid \(4x^{2}+9y^{2}=36\) and the planes \(z=0\) and \(z=1\)

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Below the paraboloid \(z=x^{2}+y^{2}\) and above the triangular region \(R \) formed by the \(x\)- and \(y\)-axes and the line \(x+y=1,\) as shown in the figure below.

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Enclosed by the cylinder \(z=9-y^{2}\) and the planes \(x=0\), \(x=4\), and \(z=0\)

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Enclosed by the surface \(z=4-x^{2}-y^{2}\) and \(z=0\)

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Enclosed by the surfaces \(x^{2}+y^{2}=1\), \(z=x\), and \(z=x^{2}\)

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The solid bounded from above by the graph of \(z=x^{2}+y^{2}\), from below by the \(xy\)-plane, and on the sides by the cylinder \(x^{2}+y^{2}=1\), as shown in the figure.

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Enclosed by a portion of the parabolic cylinder \(z=\dfrac{x^{2} }{2}\) and the four planes \(y=0\), \(y=x\), \(x=2\), and \(z=0\)

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Enclosed by the coordinate planes, the plane \(y=3\), and the surface \(z=y+1-x^{2}\)

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Enclosed by the surfaces \(z=xe^{y}\), \(z=0\), \(y=0\), \(x=1\), and \(y=x^{2}\)

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\(\int_{0}^{\sqrt{\pi /2}}\left[ \int_{y}^{\sqrt{\pi /2}}\sin x^{2}\,{\it dx}\right]\! {\it dy}\)

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\(\int_{0}^{1/2}\left[ \int_{2x}^{1}e^{y^{2}} {\it dy}\right]\! {\it dx}\)

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\(\int_{0}^{1}\left[ \int_{0}^{\sqrt{1-x^{2}}}\dfrac{1}{\sqrt{1-y^{2}}}\,{\it dy}\right]\! {\it dx}\)

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\(\int_{0}^{1}\left[ \int_{y}^{1}\dfrac{\sin x}{x}\,{\it dx}\right]{\it dy}\)

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\(\int_{0}^{1}\left[ \int_{y}^{1}\sqrt{2+x^{2}}\,{\it dx}\right]\! {\it dy}\)

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\(\int_{0}^{1}\left[ \int_{\sqrt[3]{x}}^{1}\sqrt{1+y^{4}}~{\it dy}\right]\! {\it dx}\)

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\(\int_{0}^{1}\left[ \int_{\sqrt{y}}^{1}e^{y/x}\,{\it dx}\right]\! {\it dy}\)

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\(\int_{0}^{1}\left[ \int_{\sin ^{-1}y}^{\pi /2}e^{\cos x}\,{\it dx}\right]\! {\it dy}\)

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Properties of Double Integrals Let \(R\) be the region enclosed by \(y=1\), \(y=-1\), \(x=0\), and \(x=1\); let \(R_{1}\) and \(R_{2}\) be the subregions of \(R\) in the first and fourth quadrants, respectively. Suppose \(f\) is continuous on \(R\) and \begin{eqnarray*} &&\displaystyle\iint\limits_{\kern-3ptR}3f(x,y)\,{\it dA}-2\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=\displaystyle\iint \limits_{\kern-3ptR_{2}}f(x,y)\,{\it dA}, \\[3pt] &&\displaystyle\iint\limits_{\kern-3ptR_{2}}5f(x,y)\,{\it dA}-2\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=18 \end{eqnarray*}

Find \(\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA}\).

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Properties of Double Integrals Let \(R\) be the region enclosed by \(y=0\), \(y=2\), \(x=-2\), and \(x=1\); let \(R_{1}\) and \(R_{2}\) be the subregions of \(R\) in the first and second quadrants, respectively. Suppose \(f\) is continuous on \(R\) and \begin{eqnarray*} &&\displaystyle\iint\limits_{\kern-3ptR_{2}}3f(x,y)\,{\it dA}-\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=2\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA},\\[4pt] &&\displaystyle\iint\limits_{\kern-3ptR_{1}}4f(x,y)\,{\it dA}+\displaystyle\iint\limits_{\kern-3ptR_{2}}2f(x,y)\,{\it dA}=7 \end{eqnarray*}

Find \(\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA}\).

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Properties of Double Integrals Let \(R\) be the region enclosed by \(y=x\), \(y=-1\), and \(x=2\); let \(R_{1}\) and \(R_{2}\) be the subregions of \(R\) above the \(x\)-axis and below the \(x\)-axis, respectively. Suppose \(f\) is continuous on \(R\) and \begin{eqnarray*} &&2\displaystyle\iint\limits_{\kern-3ptR_2}f(x,y)\,{\it dA}-7\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=17\\[4pt] &&\displaystyle\iint\limits_{\kern-3ptR_{2}}f(x,y)\,{\it dA}-2\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=7 \end{eqnarray*}

Find \(\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA}\).

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Properties of Double Integrals Let \(R\) be the region enclosed by \(y=x^{2}+1\), \(y=0\), \(x=-1\), and \(x=2\); let \(R_{1}\) and \(R_{2}\) be the subregions of \(R\) in the first and second quadrants, respectively. Suppose \(f\) is continuous on \(R\) and \begin{eqnarray*} &&\displaystyle\iint\limits_{\kern-3ptR_2}6f(x,y)\,{\it dA}-3\displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA}=-12\\[4pt] &&\displaystyle\iint\limits_{\kern-3ptR_{2}}4f(x,y)\,{\it dA}+2= \displaystyle\iint\limits_{\kern-3ptR_{1}}f(x,y)\,{\it dA} \end{eqnarray*}

Find \(\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA}\).

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\(\int_{0}^{1}\left[ \int_{0}^{\sqrt{1-x^{2}}}\sqrt{1-x^{2}-y^2}\,{\it dy}\right]\! {\it dx}\)

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\(\int_{0}^{2}\left[ \int_{0}^{\sqrt{4-y^{2}}}(4-x^2-y^2)\,{\it dx}\right]\! {\it dy}\)

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\(\int_{0}^{4}\left[ \int_{0}^{1}3\,{\it dy}\right]\! {\it dx}\)

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\(\int_{0}^{1}\left[ \int_{0}^{2}2\,{\it dy}\right]\! {\it dx}\)

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Find:

1. \(\displaystyle\iint\limits_{\kern-3ptR}x\,{\it dA},\) where \(R\) is the region enclosed by the circle of radius 1 centered at the origin.
2. \(4\displaystyle\iint\limits_{\kern-3ptR_{1}}x\,{\it dA},\) where \(R_{1}\) is the region in the first quadrant enclosed by the circle of radius 1 centered at the origin (This shows that although the region over which we are integrating is symmetric about the \(x\)-axis and the \(y\)-axis, we must also consider properties of the integrand on this region before we use symmetry.)

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1. Find \[ \int_{1}^{e}\left[ \int_{0}^{\ln x}\dfrac{{\it dy}}{x}\right]\! {\it dx}. \]
2. Reverse the order of integration and find the resulting iterated integral.

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1. Find \[ \int_{0}^{\pi /2}\left[ \int_{0}^{\cos x}\sin x\,{\it dy}\right]\! {\it dx}. \]
2. Reverse the order of integration and find the resulting iterated integral.

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Volume of a Liquid A tank in the shape of a half-cylinder is lying on its flat side. The radius of the tank is 1 m and its length is 4 m. If the tank is filled with liquid to a depth of \(a\) meters, what is the volume of the liquid? (Hint: Position the half cylinder as shown in the figure below.)

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Volume of a Liquid Repeat Problem 80 if the tank is buried so its rounded base is 1 m under the ground and its flat side is at ground level. See the figure below.

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Damming a Valley A dam is built in a V-shaped valley. The top of the dam is \(200\) m wide and the lowest point in the V is \(150\) m below the top, as shown in the figure. The pressure \(P\) due to the water at a depth \(d\) meters below the surface is \(P=\rho gd\), where \(\rho =1000 \text{kg}/ \text{m}^{3}\) and \(g=9.8 \text{m}/ {\text s}^{2}\). What is the total outward force \(F\) from the water that this dam must be able to withstand? Express your answer in both newtons and pounds using the fact that \(1 {\text N} \approx 0.2248 \text{lb}\). (Recall that pressure is \(P=\dfrac{F}{A},\) so the force \(F=PA.)\)

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Explain if the following equality is true or false without finding the exact value of the two integrals: \[ \int_{-3}^{3}\int_{-\sqrt{ 9-y^{2}}}^{\sqrt{9-y^{2}}} ( 7-x)\,{\it dx}\, {\it dy}=4\int_{0}^{3}\int_{0}^{ \sqrt{9-y^{2}}}( 7-x)\,{\it dx}\, {\it dy} \]

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Volume Find the volume of the tetrahedron enclosed by the coordinate planes and the plane \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\). (Assume that \(a,b\), and \(c\) are positive.)

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Find \(\displaystyle\iint\limits_{\kern-3ptR}xy\,{\it dA}\), where \(R\) is the region enclosed by \(y=4-x^{2}\) and \(y=x^{2}\) to the right of the \(y\)-axis.

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Find \(\displaystyle\iint\limits_{\kern-3ptR}xy\,{\it dA}\), where \(R\) is the region enclosed by \(x=4-y^{2}\) and \(x=y^{2}\) above the \(x\)-axis.

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Find \(\displaystyle\iint\limits_{\kern-3ptR}e^{\sqrt{x}}\,{\it dA}\), where \(R\) is the dregion enclosed by \(y=x\), \(x=1\), \(y=0\).

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1. Graph the region \(R\) defined by \(0\leq a\leq y\leq b,\) \(0\leq a\leq x\leq y.\)
2. If \(z=f(x)\) is continuous on \(R,\) show that \( \displaystyle\iint\limits_{\kern-3ptR}f(x) \,{\it dA}=\int_{a}^{b}(b-x)f(x)\,{\it dx}\).

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Properties of Double Integrals Using properties of double integrals, show that if the functions \(f\) and \(g\) are integrable on \(R\) and \( f(x,y)\leq g(x,y)\) for all \((x,y)\) in \(R\), then \(\displaystyle\iint\limits_{\kern-3ptR}f(x,y)\,{\it dA} \leq \displaystyle\iint\limits_{\kern-3ptR}g(x,y)\,{\it dA}\).

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Volume Find the volume of the solid under the graph of the elliptic paraboloid \(z=8-2x^{2}-y^{2}\) and above the first quadrant of the \(xy\)-plane. (Hint: Use Wallis’s formula from Section 7.1, Problem 69.)