Question

True or False A solid \(E\) in space is called \(xy\)-simple if it is enclosed on the top by a surface \(z=z_{2}(x, y)\) and on the bottom by a surface \(z=z_{1}(x, y)\), where \(z_{1}\leq z_{2}\) and \( z_{1}\) and \(z_{2}\) are continuous, and the sides of \(E\) are a cylinder whose intersection with the \(xy\)-plane forms a closed, bounded region \(R.\)

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True or False The region \(R\) resulting from the projection onto the \(xy\)-plane of an \(xy\)-simple solid \(E\) is both \(x\)-simple and \(y\)-simple.

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True or False The mass \(M\) of a solid with a mass density \(\rho =\rho (x, y, z)\) that is continuous in a region \(E\) of volume \(V\) is given by \(M=\iiint\limits_{\kern-3ptE}\rho (x, y, z) \,{\it dV}.\)

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True or False The triple integral \( \iiint\limits_{\kern-3ptE}\,{\it dV}\) can be interpreted geometrically as the area of the region \(R\) that is the projection of \(E\) onto the \(xy\)-plane.

Question

\(\iiint\limits_{\kern-3ptE}xy^{2}\,{\it dV}\), \(E\) is the closed box \(0\leq x\leq 1\), \(0\leq y\leq 1\), \(0\leq z\leq 4\).

Question

\(\iiint\limits_{\kern-3ptE}x^{2}yz\,{\it dV},\) \(E\) is the closed box \( 0\leq x\leq 2\), \(0\leq y\leq 1\), \(0\leq z\leq 2\).

Question

\(\iiint\limits_{\kern-3ptE}(x^{2}+y^{2}+z^{2})\,{\it dV},\) \(E\) is the closed box \(0\leq x\leq 1\), \(0\leq y\leq 1\), \(0\leq z\leq 1\).

Question

\(\iiint\limits_{\kern-3ptE}(x^{2}-y^{2}+z^{2})\,{\it dV},\) \(E\) is the closed box \(0\leq x\leq 1\), \(0\leq y\leq 2\), \(0\leq z\leq 1\).

Question

\(\iiint\limits_{\kern-3ptE}e^{z}\sin x\cos y\,{\it dV},\) \(E\) is the closed box \(0\leq x\leq \dfrac{\pi }{2}\), \(0\leq y\leq \dfrac{\pi }{2}\), \( 0\leq z\leq 1\).

Question

\(\iiint\limits_{\kern-3ptE}e^{-z}\cos x\cos y\,{\it dV},\) \(E\) is the closed box \(0\leq x\leq \dfrac{\pi }{2}\), \(0\leq y\leq \dfrac{\pi }{2}\), \(0\leq z\leq 1\).

Question

\(\int_{0}^{2}\int_{0}^{2-3x}\int_{0}^{x+y}x\,{\it dz}\,{\it dy}\,{\it dx}\)

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\(\int_{0}^{1}\int_{0}^{-x}\int_{0}^{2x+y}z\,{\it dz}\,{\it dy}\,{\it dx}\)

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\(\int_{0}^{3}\int_{z}^{z+2}\int_{y}^{y+z}( 2x+1) \,{\it dx}\,{\it dy}\,{\it dz}\)

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\(\int_{0}^{2}\int_{y}^{y+2} \int_{y}^{x+y}( 4z-1) \,{\it dz}\,{\it dx}\,{\it dy}\)

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\(\int_{0}^{2}\int_{0}^{1}\int_{0}^{y}e^{x}\,{\it dx}\,{\it dy}\,{\it dz}\)

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\(\int_{1}^{2}\int_{0}^{z}\int_{0}^{\sqrt{y}}xe^{x^{2}}\,{\it dx}\,{\it dy}\,{\it dz}\)

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\(\int_{0}^{\pi /2}\int_{y}^{\pi /2}\int_{0}^{xy}\sin \dfrac{z }{y}\,{\it dz}\,{\it dx}\,{\it dy}\)

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\(\int_{0}^{\pi/2}\int_{x}^{\pi /2}\int_{0}^{xy}\cos \dfrac{z}{x}\,{\it dz}\,{\it dy}\,{\it dx}\)

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\(E\) is the solid enclosed by the surfaces \(z=0\) and \(z=5-x-y\) , whose projection onto the \(xy\)-plane is the region enclosed by the rectangle \(0\leq x\leq 1\) and \(0\leq y\leq 3\).

Question

\(E\) is the solid enclosed by the surfaces \(z=0\) and \( z=16-x^{2}-y^{2}\), whose projection onto the \(xy\)-plane is the region enclosed by the rectangle \(0\leq x\leq 2\) and \(0\leq y\leq 1\).

Question

\(E\) is the solid enclosed by the surfaces \(z=0\) and \(z=xy\), whose projection onto the \(xy\)-plane is the region enclosed by the triangle with vertices \((0, 0, 0)\), \((0, 1, 0)\), and \((1, 0, 0)\).

Question

\(E\) is the solid enclosed by the surfaces \(z=0\) and \( z=x^{2}+y^{2}\), whose projection onto the \(xy\)-plane is the region enclosed by the triangle with vertices \((0, 0, 0)\), \((1, 0, 0)\), and \((0, 2, 0)\).

Question

\(E\) is the solid enclosed by the surfaces \(z=0\) and \(z=3-x-y\) , whose projection onto the \(xy\)-plane is the region enclosed by the triangle with vertices \(( 0, 0, 0) ,\) \(( 1, 1, 0) \), and \( (1, -1, 0) \).

Question

\(E\) is the solid enclosed by the surfaces \(z=0\) and \(z=3+2y\), whose projection onto the \(xy\)-plane is the region enclosed by the triangle with vertices \(( -1, 0, 0) , ( 0, 1, 0) ,\) and \(( 2, 0, 0) .\)

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\(\int_{0}^{1}\int_{1}^{x^{2}}\int_{0}^{\sqrt{x} }ye^{x}\, {\it dz}\, {\it dy}\,{\it dx}\)

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\(\int_{3}^{4}\int_{2}^{3} \int_{0}^{1}\ln ( x^{2}+y^{2}+z^{2}) \,{\it dx}\, {\it dy}\, {\it dz}\)

Question

\(\iiint\limits_{\kern-3ptE}x\,{\it dV},\) if \(E\) is the solid enclosed by the tetrahedron having vertices at \((0, 0, 0)\), \((1, 1, 0)\), \((1, 0, 0)\), \((1, 0, 1)\).

Question

\(\iiint\limits_{\kern-3ptE}(x^{2}+z^{2})\,{\it dV}\), if \(E\) is the solid enclosed by the tetrahedron having vertices at \((0, 0, 0)\), \((1, 1, 0)\), \( (1, 0, 0) \), \((1, 0, 1)\).

Question

\(\iiint\limits_{\kern-3ptE}(xy+3y)\,{\it dV}\), if \(E\) is the solid enclosed by the cylinder \(x^{2}+y^{2}=9\) and the planes \(x+z=3\), \(y=0\), and \(z=0\).

Question

\(\iiint\limits_{\kern-3ptE}xyz\,{\it dV}\), if \(E\) is the solid enclosed by the cylinders \(x^{2}+y^{2}=1\) and \(x^{2}+z^{2}=1\). See the figure.

Question

\(\int_{-2}^{2}\int_{0}^{\sqrt{4-y^{2}}}\int_{0}^{1}\,{\it dz}\,{\it dx}\, {\it dy}\)

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\(\int_{0}^{2}\int_{0}^{\sqrt{4-y^{2}}}\int_{0}^{4-x^{2}-y^{2}}\,{\it dz}\,{\it dx}\, {\it dy}\)

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\(\int_{0}^{1}\int_{0}^{x^{2}}\int_{0}^{y}\,{\it dz}\,{\it dy}\,{\it dx}\)

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\(\int_{0}^{1}\int_{y^{2}}^{1}\int_{0}^{1-x}\,{\it dz}\,{\it dx}\,{\it dy}\)

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Volume of a Solid Find the volume \(V\) of the solid enclosed by \(y^{2}=z\), \(x=0\), and \(x=y-z.\)

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Volume of a Solid Find the volume \(V\) of the solid enclosed by \(z=4-y^{2},\) \(z=9-x\), \(x=0\), and \(z=0.\)

Question

Set up, but do not find, \(\iiint\limits_{\kern-3ptE}xy\,{\it dV},\) where \(E\) is the solid enclosed by the surfaces \(z=1-x-y\) and \(z=3-x-y\), whose projection onto the \(xy\)-plane is the region enclosed by the circle \( x^{2}+y^{2}=1\).

Question

Set up, but do not find, \(\iiint\limits_{\kern-3ptE}xy~{\it dV},\) where \(E\) is the solid enclosed by the surfaces \(z=0\) and \(z=x^{2}+y\), whose projection onto the \(xy\)-plane is the region enclosed by the circle \( x^{2}+y^{2}=4\).

Question

Enclosed by \(z=x^{2}+y^{2}\) and \(z=16-x^{2}-y^{2}\).

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Enclosed by \(z=x^{2}+y^{2}\) and \(z=2-x\).

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Enclosed by \(z^{2}=4x\) and \(x^{2}+y^{2}=2x.\)

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Volume

1. Find the volume \(V\) of the solid enclosed by \(x^{2}+2y^{2}=1\) and the planes \(z=x+1\) and \(z=4.\)
2. Graph the solid.

Question

Mass Find the mass \(M\) of an object in the shape of a cube of edge \(a\) if its mass density is proportional to the square of the distance from one corner.

Question

Mass Set up, but do not find, the integral that equals the mass \(M\) of an object in the shape of a right circular cylinder of height \(h\) and radius \(a\), if its mass density is proportional to the square of the distance from the axis of the cylinder.

Question

Mass Set up, but do not find, the integral that equals the mass \(M\) of an object in the shape of a tetrahedron cut from the first octant by the plane \(x+y+z=1\), as shown in the figure, if its mass density is proportional to the product of the distances from the three coordinate planes.

Question

Mass A cylindrical bar of radius \(R\) and length \(2L\) is positioned with its axis along the \(x\)-axis and its center of mass at the origin. The mass density of the bar is given by \(\rho ( x, y, z) =kz^{2}. \) Show that the mass \(M\) of the bar is \(M=\dfrac{1}{2}\pi kR^{4}L.\)

Question

Moments of Inertia Set up the integrals that equal the moments of inertia \(I_{x}\) and \(I_{y}\) for the solid region enclosed by the hemisphere \(z=\sqrt{9-x^{2}-y^{2}}\) and the \(xy\)-plane, if the mass density is proportional to the distance from the \(xy\)-plane.

Question

Set up, but do not find, the integral of the function \( f(x, y, z)=x^{2}yz\) over the solid enclosed by the cone \(3x^{2}+3y^{2}=z^{2}\), \(z\geq 0\), and the plane \(z=3\).

Question

\(E\) is the solid enclosed by the coordinate planes and the plane \(x+2y+3z=6\).

Question

\(E\) is the solid enclosed by the coordinate planes and the plane \(x+y+z=3\).

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Volume of an Ellipsoid Show that the volume of the ellipsoid \( \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) is \(\dfrac{ 4}{3}\pi abc\). (Assume that \(a\), \(b\), and \(c\) are positive.) What does this formula reduce to if \(a=b=c\)?

Question

Volume of a Solid Find the volume of the solid in the first octant enclosed by the coordinate planes, \(a^{2}y=b(a^{2}-x^{2})\) and \( a^{2}z=c(a^{2}-x^{2})\), \(a\gt0\), \(b\gt0\), \(c\gt0\).

Question

Volume of a Solid Find the volume \(V\) of the region enclosed by \(z=0,\) \(z=1-x^{2},\) and \(z=1-y^{2}\)

Question

Average Value of a Function The average value of \(f\) over a region \(R\) that is not necessarily rectangular is defined to be the number \( \dfrac{1}{A}\displaystyle\iint\limits_{\kern-3ptR} f(x,y)\,{\it dA}\).

1. In single variable calculus the average value of a function \(f\) over the closed interval \([a,b] \) is defined to be the number \(\dfrac{ 1}{b-a}\int_{a}^{b}f(x)\,{\it dx}\). In what sense is this a special case of the above definition of the average value of \(f\) over \(R\)?
2. Let \(w=\) \(f(x, y, z)\) be integrable over a solid \(E\) in space. What definition would you give for the average value of \(f\) over \(E\)?

Question

Show that the following integrals represent the same volume. Do not find the integrals.

1. \(4\int_{0}^{4}\int_{0}^{\sqrt{16-x^{2}} }\int_{(x^{2}+y^{2})/4}^{4}\,{\it dz}\,{\it dy}\,{\it dx}\)
2. \(4\int_{0}^{4}\int_{0}^{2\sqrt{z}}\int_{0}^{\sqrt{4z-x^{2}} }\,{\it dy}\,{\it dx}\,{\it dz}\)
3. \(4\int_{0}^{4}\int_{y^{2}/4}^{4}\int_{0}^{\sqrt{4z-y^{2}} }\,{\it dx}\,{\it dz}\,{\it dy}\)

Question

Show that: \[ \int_{a}^{b}\int_{a}^{z}\int_{a}^{y}f(x)\,{\it dx}\,{\it dy}\,{\it dz}=\int_{a}^{b}\frac{ (b-x)^{2}}{2}f(x)\,{\it dx} \]