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True or False  When a smooth curve \(C\) represented by the parametric equations \(x=x(t) ,\) \(y=y(t) ,\) \( y\geq 0,\) \(a\leq t\leq b,\) is revolved about the \(x\)-axis, the surface area \( S\) of the solid of revolution is given by \(S=2\pi \int_{a}^{b}x(t) \sqrt{\left( \dfrac{dx}{dt}\right) ^{2}+\left( \dfrac{dy}{dt} \right) ^{2}}dt.\)

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The surface area \(S\) of a solid of revolution generated by revolving the smooth curve \(C\) represented by \(x=x(t),\;y=y(t),\;a\leq t\leq b\), where \(x(t)\geq 0\), about the \(y\)-axis is \(S=\) ______________.

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\(x(t) =3t^{2}, \, y(t) =6t;\;0\leq t\leq 1\)

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\(x(t) =t^{2}, \, y(t) =2t;\;0\leq t\leq 3 \)

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\(x( \theta) =\cos ^{3}\theta, \, y( \theta) =\sin ^{3}\theta ;\; 0\leq \theta \leq \dfrac{\pi }{2}\)

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\(x(t) = t-\sin t, \ \, y(t) = 1-\cos t;\;0\leq t\leq \pi \)

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\(y=x^{3},\;0\leq x\leq 1\)

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\(y=4x^{3},\;0\leq x\leq 2\)

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\(y=\dfrac{x^{4}}{8}+\dfrac{1}{4x^{2}},\;1\leq x\leq 2\)

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\(y=\sqrt{x};\;1\leq x\leq 9\)

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\(y=e^{x},\;0\leq x\leq 1\)

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\(y=e^{-x},\;0\leq x\leq 1\)

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\(y= \sqrt{a^{2}-x^{2}},\;-a\leq x\leq a\)

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\(y=\dfrac{a}{2}(e^{x/a}+e^{-x/a}),\;0\leq x\leq a\)

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\(x(t) =3t^{2}, y(t) =2t^{3};\;0\leq t\leq 1\)

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\(x(t) =2t+1, \ y(t) =t^{2}+3;\;0\leq t\leq 3\)

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\(x(t) =2\sin t, y(t) =2\cos t;\;0\leq t\leq \dfrac{\pi }{2}\)

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\(x(t) =3\cos t, y(t) =2\sin t;\;0\leq t\leq \dfrac{\pi }{2} \)

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\(x=\dfrac{1}{4}y^{2},\;0\leq y\leq 2\)

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\(x^{2/3}+y^{2/3}=a^{2/3};\;x\geq 0, 0\leq y\leq a\)

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Find the surface area of the solid generated by revolving one arch of the cycloid \(x(t) =6( t-\sin t) ,\) \(y(t) =6( 1-\cos t) \) about the \(x\) -axis.

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Find the surface area of the solid generated by revolving the graph of \(y=\ln x,\) \(1\leq x\leq 10,\) about the \(x\)-axis.

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Gabriel’s Horn  The surface formed by revolving the region between the graph of \(y=\dfrac{1}{x},\) \(x\geq 1,\) and the \(x\)-axis about the \(x\)-axis is called Gabriel’s horn. See the figure.

1. Find the surface area of Gabriel’s horn.
2. Find the volume of Gabriel’s horn.

Interesting Note:  The volume of Gabriel’s horn is finite, but the surface area of Gabriel’s horn is infinite.

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Surface Area  Find the surface area of the solid of revolution obtained by revolving the graph of \(y=e^{-x}\), \(x\geq 0\), about the \(x\)-axis.

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Surface Area of a Catenoid  When an arc of a catenary \( y=\cosh x,\) \(a\leq x\leq b\), is revolved about the \(x\)-axis, it generates a surface called a catenoid, which has the least surface area of all surfaces generated by rotating curves having the same endpoints. Find its surface area. See the figure.

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Surface Area of a Sphere  Find a formula for the surface area of a sphere of radius \(R\).

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Surface Area  Show that the surface area \(S\) of a right circular cone of altitude \(h\) and radius \(b\) is \(S=\pi b \sqrt{h^{2}+b^{2}}.\)

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Searchlight  The reflector of a searchlight is formed by revolving an arc of a parabola about its axis. Find the surface area of the reflector if it measures \(1\) m across its widest point and is \(\dfrac{1 }{4}\) m deep.

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Surface Area of a Bead  A sphere of radius \(R\) has a hole of radius \(a<R\) drilled through its center. The axis of the hole coincides with a diameter of the sphere. Find the surface area of the part of the sphere that remains.

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Surface Area of a Plug  A plug is made to repair the hole in the sphere in Problem 29. What is the surface area of the plug?