Question

True or False Suppose \(R\) is a closed, rectangular region in the \(xy\)-plane with sides of length \(\Delta x\) and \( \Delta y\). If the perimeter of \(R\) is projected upward parallel to the \(z\) -axis, the result is a cylinder. If \(S_{p}\) is the area cut from the plane \( z=ax+by+c\) by this cylinder, then \(S_{p}=\sqrt{a^{2}+b^{2}+1}\,\Delta x\Delta y\).

Question

True or False Suppose \(z=f(x,y)\) is a function of two variables defined on a closed, bounded region \(R\). If \(f_{x}\) and \(f_{y}\) are continuous on \(R\), then the area \(S\) of the part of the surface that lies over \(R\) is given by \(S=\displaystyle\iint\limits_{\kern-3ptR}\sqrt{f_{x}(x,y)+f_{y}(x,y)+1 }d\!A\).

Question

Find the surface area of the part of the plane \(2x+2y+z=6\) that lies above the region in the \(xy\)-plane bounded by \(x^{2}+y^{2}=4,\) the \(x\)-axis, and the \(y\)-axis, as shown in the figure.

Question

Find the surface area of the part of the plane \(4z-2x-y=8\) that lies over the region enclosed by the triangle with vertices \((0,0,0)\), \((1,0,0)\), and \(( 1,1,0)\), as shown in the figure.

Question

The part of the surface \(z=f(x,y) =\sqrt{ 9-x^{2}}\) that lies above the rectangle enclosed by the lines \(x=0,\) \(x=2,\) \( y=-1,\) \(y=4\)

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The part of the surface \(z=f(x,y) =\sqrt{8-y^{2}}\) that lies above the rectangle enclosed by the lines \(x=0,\) \(x=5,\) \(y=0,\) \( y=2\)

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The part of the surface \(z=\dfrac{2}{3}( x^{3/2}+y^{3/2})\) that lies above the triangle enclosed by the lines \( x=0\), \(y=0\), and \(2x+3y=6\)

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The part of the surface \(z=\dfrac{2}{3}(x^{3/2}+y^{3/2})\) that lies above the triangle enclosed by \(x=0\), \(y=0\), and \(3x+y=3\)

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The part of the paraboloid \(z=4-x^{2}-y^{2}\) that lies above the \(xy\)-plane

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The part of the cylinder \(z=\sqrt{a^{2}-x^{2}}\) that lies above the square defined by \(-\dfrac{1}{2}a\leq x\leq \dfrac{1}{2}a\) and \(-\dfrac{1 }{2}a\leq y\leq \dfrac{1}{2}a\), \(a\gt 0\)

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The part of the cone \(z=\sqrt{x^{2}+y^{2}}\) that lies inside the cylinder \(x^{2}+y^{2}=2x\)

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The part of the sphere \(x^{2}+y^{2}+z^{2}=4z\) that lies within the paraboloid \(x^{2}+y^{2}=2z\)

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The part of the surface \(z=xy\) in the first octant that lies within the cylinder \(x^{2}+y^{2}=a^{2}\), \(a\gt 0\)

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The part of the surface \(z=x^{2}-y^{2}\) in the first octant that lies within the cylinder \(x^{2}+y^{2}=4\)

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The part of the sphere \(x^2+y^{2}+z^{2}=4z\) that lies between the planes \(z=1\) and \(z=3\).

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The part of the sphere \(x^{2}+y^{2}+z^{2}=4a^{2}\) that lies inside the cylinder \(x^{2}+y^{2}=2ax\), \(a\gt 0\)

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Surface Area Find the surface area cut from the hyperbolic paraboloid \(y^{2}-x^{2}=6z\) by the cylinder \(x^{2}+y^{2}=36\).

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Surface Area Find the surface area of the part of \(z=4-y^{2}\) that lies in the first octant and is enclosed by \(z=0\), \(x=0\), \(x=y\), and \( y=2\).

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Surface Area Find the surface area of the part of \(x^{2}=y\) that lies in the first octant under the plane \(x+z=3\). (Hint: Project the surface onto the \(xz\)-plane.)

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Surface Area Find the surface area of the part of the paraboloid \(z=9-x^{2}-y^{2}\) that lies between the planes \(z=0\) and \(z=8\).

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Surface Area Find the surface area of the part of the hemisphere \(x^{2}+y^{2}+z^{2}=16,\) \(z\geq 0,\) that lies inside the cylinder bounded by \(x^{2}+y^{2}=4.\) See the figure.

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Surface Area of the Pantheon The Pantheon in Rome is the largest unreinforced concrete dome in the world. Its inner surface is a hemisphere of diameter \(43.3 \text m\) with an open circular ocular \(9.1 \text m\) in diameter cut from its apex. Find the inner surface area of the dome.

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Surface Area

1. Graph the surface \(z=10e^{-( x^{2}+y^{2})}\) that lies inside the cylinder \(x^{2}+y^{2}=4\).
2. Find the surface area of the surface graphed in (a).

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Surface Area

1. Set up, but do not evaluate, the integral to find the surface area of the part of the surface \(z=1-x^{4}-y^{2} \) that lies above the \(2xy\)-plane.
2. Graph the surface.

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Derive the formula \(S=\pi a\sqrt{a^{2}+h^{2}}\) for the lateral surface area of a right circular cone of base radius \(a\) and altitude \(h\).

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Show that the area of the first-octant portion of the plane \( \dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\) (where \(a, b\), and \(c\) are positive) is \(S=\dfrac{1}{2}\sqrt{a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}}\).

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Use a double integral to derive the formula for the surface area of a sphere of radius \(R\).

Question

Let \(F\) denote a function of three variables that possesses continuous first-order partial derivatives at each point of its domain. Suppose also that \(F_{z}(x,y,z)\) is never \(0\). If \(S\) is the area of the part of the surface \(F(x,y,z)=0\) that lies over the closed, bounded region \( R\), show that \[ \begin{array}{rcl} S=\int\limits\int_{\kern-11ptR}\dfrac{\sqrt{[F_{x}(x,y,z)]^{2}+[F_{y}(x,y,z)]^{2}+[F_{z}(x,y,z)]^{2}}}{|F_{z}(x,y,z)|}\,d\!A \end{array} \]

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Question

Surface Area The center of a sphere of radius \(R\) is on the surface of a right circular cylinder of base radius \(\dfrac{R}{2}\). Find the surface area of the sphere inside the cylinder. See the figure.

Question

For the plane surface \(F(x,y,z)=Ax+By+Cz-D=0\), \(C\neq 0\), if \( z=f(x,y)\), show that the surface area \(S=\displaystyle\iint\limits_{\kern-3ptR}\sqrt{ [f_{x}(x,y)]^{2}+[f_{y}(x,y)]^{2}+1}\,d\!A\) can be written as \[ S=\displaystyle\iint\limits_{\kern-3ptR}\sec \gamma \,d\!A \]

where \(\gamma \) is the positive acute angle between the normal \(\mathbf{n}=A \mathbf{i}+B\mathbf{j}+C\mathbf{k}\) to the plane and \(\mathbf{k}\).