Question

VolumesFind the volume of the solid generated by revolving the region bounded on the left by the parabola  x = y² + 1 and on the right by the line x = 5  aboutc.  the line x = 5

Accepted Answer

First, identify the region bounded by the curves: the parabola given by $$x = y$$^{2}$$ + 1 on $$the left and the vertical line $$x = 5 on $$the right. Determine the range of $$y by $$finding the points where these two curves intersect. Set $$y^{2} + 1 = 5$ and solve for y$. Since the solid is generated by revolving the region about the line x = 5$, consider the method of cylindrical shells or washers. Here, the axis of rotation is vertical, so using the shell method with respect to y$ is convenient. For the shell method, the radius of a shell at a given y$ is the distance from the line x = 5$ to the shell, which is 5 - x$. Since x$ varies from x$$ = $$y^{2} + 1$ to x = 5$, the shell radius is 5$$ - $$y^{2} - 1 = 4$$ - $$y^{2}. $$The height of the shell is the vertical thickness, which corresponds to the difference in $$x$$ values, but since we revolve around $$x=5$$, the height is the length along $$x$$ between the parabola and the line, which is $$5 - (y$$^{2}$$ + 1) = 4 - y$$^{2}$$. Set up the volume integral using the shell method formula: V = 2$$\pi \int_{a}^{b} (	ext{radius})(	ext{height}) \, dy$$. Here, the radius is the distance from the axis of rotation to the shell, and the height is the length of the shell. Determine the limits of integration a$ and b$ from the intersection points found in step 1. Write the integral explicitly as V = 2$$\pi \int_{-c}^{c} (	ext{radius})(	ext{height}) \, dy$$, where c$ is the positive y$-value of the intersection. Substitute the expressions for radius and height, simplify the integrand, and prepare to evaluate the integral to find the volume.

Volumes
Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about
c. the line x = 5

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Key Concepts

Volume of Solids of Revolution

Shell Method

Setting up the Limits of Integration and Radius/Height Expressions