Now You Try

<p> Find the volume enclosed by the surface that is generated by revolving the equilateral triangle with vertices <code class='latex inline'>(0, 0), (a, 0), (\frac{1}{2}a, \frac{1}{2}\sqrt{3}a) </code> about the x-axis. </p>

<p>Use the method of cylindrical shells to nd the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.</p><p><code class='latex inline'>
\displaystyle
x = -3y^2 + 12y -9, x = 0
</code></p>

<p>Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.</p><p><code class='latex inline'>\displaystyle
y=x^3, y = 8, x= 0;
</code> about <code class='latex inline'>x = 3</code></p>

<p>The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.</p><p><code class='latex inline'>\displaystyle
x^2 +(y-1)^2 =1;
</code> about the y-axis.</p>

<p>Use the method of cylindrical shells to nd the volume generated by rotating the region bounded by the given curves about the y-axis.</p><p><code class='latex inline'>
\displaystyle
y=e^{-x^2}, y = 0, x = 0, x = 1
</code></p>

<p>Let <code class='latex inline'>S</code> be the solid obtained by rotating the region shown in the figure about the y-axis. Explain why it is awkward to use slicing to find the volume <code class='latex inline'>V</code> of <code class='latex inline'>S</code>. Sketch a typical approximating shell. What are its circumference and height? Use shells to find <code class='latex inline'>V</code>.</p><img src="/qimages/5413" />

<p>Use the method of cylindrical shells to nd the volume generated by rotating the region bounded by the given curves about the y-axis.</p><p><code class='latex inline'>
\displaystyle
y = x^2, y= 6x -2x^2
</code></p>

<p>The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.</p><p><code class='latex inline'>\displaystyle
x=(y-1)^2, x -y =1;
</code>, <code class='latex inline'>x = -1</code></p>

<p> The region between the graph of <code class='latex inline'>f(x) = \sqrt{x}</code> and the x-axis for <code class='latex inline'>0 \leq x \leq 4</code> is revolved about the line <code class='latex inline'>y = 2</code>. Find the volume of the solid that is generated.</p>

<p>Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region and a typical shell.</p><p><code class='latex inline'>
\displaystyle
y = 4x^2, 2x + y = 6
</code></p>

<p>Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.</p><p><code class='latex inline'>\displaystyle
x = 2y^2, y \geq 0, x = 2;
</code> about <code class='latex inline'>y = -2</code></p>

<p>A base of a solid is a circle with radius <code class='latex inline'>a</code>, and every plane section perpendicular to a diameter is a square. Find the volume of the solid in terms of <code class='latex inline'>a</code>.</p>

<p>Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.</p><p><code class='latex inline'>\displaystyle
y =4x -x^2, y =3;
</code> about <code class='latex inline'>x = 1</code></p>

<p>The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.</p><p><code class='latex inline'>\displaystyle
x = (y-3)^2, x = 4
</code>, <code class='latex inline'>y = 1</code></p>

<p>Use the method of cylindrical shells to nd the volume generated by rotating the region bounded by the given curves about the y-axis.</p><p><code class='latex inline'>
\displaystyle
y=x^3, y = 0, x= 1, x = 2
</code></p>

<p>The base of a solid is region bounded by <code class='latex inline'>y = e^{-x}</code>, the x-axis, the y-axis, and the line <code class='latex inline'>x = 1</code>. Each cross section perpendicular to the x-axis is a square. Find the volume of the solid that is generated.</p>

<p>The integral represents the volume of a solid. Describe the solid.</p><p><code class='latex inline'>\displaystyle
\int^{1}_{0} 2\pi (2-x)(3^x -2^x) dx
</code></p>

<p>Let <code class='latex inline'>V</code> be the volume of the solid obtained by rotating about the y-axis the region bounded by <code class='latex inline'>y=\sqrt{x}</code> and <code class='latex inline'>y=x^2</code>. Find <code class='latex inline'>V</code> both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.</p>

<p>The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.</p><p><code class='latex inline'>\displaystyle
y = -x^2 + 6x -8, y =0;
</code> about the x-axis</p>

<p>Use the method of cylindrical shells to nd the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.</p><p><code class='latex inline'>
\displaystyle
xy =1, x = 0, y = 1, y =3
</code></p>

<p>The integral represents the volume of a solid. Describe the solid.</p><p><code class='latex inline'>\displaystyle
2\pi \int^{4}_{1} \frac{y + 2}{y^2}dy
</code></p>

<p>Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the given line.. Sketch the region and a typical shell.</p><p><code class='latex inline'>
\displaystyle
y = x^2, y = 0, x = 1, x = 2
</code> rotate about <code class='latex inline'>x = 4</code>.</p>

<p>The integral represents the volume of a solid. Describe the solid.</p><p><code class='latex inline'>\displaystyle
\int^{3}_{1}2\pi y\ln y dy
</code></p>

<p>Use the method of cylindrical shells to nd the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.</p><p><code class='latex inline'>x + y = 4, x = y^2 -4y + 4</code></p>

<p>Use the method of cylindrical shells to nd the volume generated by rotating the region bounded by the given curves about the y-axis.</p><p><code class='latex inline'>
\displaystyle
y=\sqrt[3]{x}, y =0, x =1
</code></p>

<p>Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.</p><p><code class='latex inline'>\displaystyle
y = \sqrt{x}, x = 2y;
</code> about <code class='latex inline'>x = 5</code></p>

<p>Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.</p><p><code class='latex inline'>\displaystyle
y = 4-2x, y = 0, x = 0;
</code> about <code class='latex inline'>x = -1</code></p>

<p>Let <code class='latex inline'>S</code> be the solid obtained by rotating the region shown in the figure about the y-axis. Explain why it is awkward to use slicing to find the volume <code class='latex inline'>V</code> of <code class='latex inline'>S</code>. Sketch a typical approximating shell. What are its circumference and height? Use shells to find <code class='latex inline'>V</code>.</p><img src="/qimages/5414" />

<p>Use the method of cylindrical shells to nd the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.</p><p><code class='latex inline'>
\displaystyle
y = x^{\frac{3}{2}}, y=8, x = 0
</code></p>

<p>Use the method of cylindrical shells to nd the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.</p><p><code class='latex inline'>
\displaystyle
y =\sqrt{x}, x = 0, y = 2
</code></p>

<p>Use the method of cylindrical shells to nd the volume generated by rotating the region bounded by the given curves about the y-axis.</p><p><code class='latex inline'>
\displaystyle
y = 4x-x^2, y = x
</code></p>

<p>Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.</p><p><code class='latex inline'>\displaystyle
x = 2y^2, y \geq 0, x =2
</code> about <code class='latex inline'>y = 2</code></p>

<p>The integral represents the volume of a solid. Describe the solid.</p><p><code class='latex inline'>\displaystyle
\int^{3}_{0}2\pi x^5 dx
</code></p>

<p>The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.</p><p><code class='latex inline'>\displaystyle
y^2-x^2=1, y=2;
</code> about the y-axis.</p>

<p>The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.</p><p><code class='latex inline'>\displaystyle
y^2-x^2=1, y=2
</code> about the x-axis</p>

<p>Use the method of cylindrical shells to nd the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.</p><p><code class='latex inline'>
\displaystyle
x = 1+(y-2)^2, x = 2
</code></p>

<p>The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.</p><p><code class='latex inline'>\displaystyle
y =-x^2 +6x -8, y=0;
</code> about the y-axis</p>