Question

VolumesFind the volume of the solid generated by revolving the region bounded by the parabola  y² = 4x and the line y = x aboutd. the line y = 4

Accepted Answer

First, identify the region bounded by the curves. The parabola is given by $$y^{2} = 4x$, which can be rewritten as x$$ = \frac{$$y^{2}$$}{4}$$. The line is y = x$. To find the limits of integration, find the points of intersection by solving y = x$ and y$$^{2}$$ = 4x$$ simultaneously. Set $$y = x$$ into the parabola equation: $$y^{2} = 4y$. This simplifies to y$$^{2}$$ - 4y = 0$$, or $$y(y - 4) = 0. So$$, the intersection points are at $$y = 0$$ and $$y = 4. $$These will be the bounds for integration. Since the solid is generated by revolving the region about the line $$y = 4$$, use the method of washers (disks with holes). For a given $$y$$ between 0 and 4, the outer radius $$R(y)$$ and inner radius $$r(y)$$ are the distances from the line $$y=4 to $$the curves. Calculate the outer radius $$R(y) as $$the distance from $$y=4 to $$the curve closer to the axis of revolution. The outer radius corresponds to the line $$y = x$$, so $$R(y) = 4 - y. $$The inner radius corresponds to the parabola $$x = \frac{y$$^{2}$$}{4}$$, so express $$x in $$terms of $$y$$ and find the horizontal distance to the axis of revolution. Since the axis is horizontal, the radius is vertical distance, so for the parabola, the radius is $$4 - y as $$well, but we need to consider the $$x-$$values to find the horizontal boundaries. Because the axis of revolution is horizontal ($$y=4$$), and the region is bounded between the curves in the $$xy-$$plane, it is easier to express $$x in $$terms of $$y$$ and use the shell method. The shell radius is the distance from $$y to$$ $$4$$, which is $$4 - y$$, and the shell height is the horizontal distance between the curves, which is $$x_{right} - x_{left} = y - \frac{y$$^{2}$$}{4}. $$Set up the volume integral using the shell method: $$V = 2\pi \int_{0}^{4} (4 - y) \left(y - \frac{y$$^{2}$$}{4}\right) dy.$$

Volumes
Find the volume of the solid generated by revolving the region bounded by the parabola y² = 4x and the line y = x about
d. the line y = 4

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Key Concepts

Setting up the region bounded by curves

Method of cylindrical shells or washers for volumes of revolution

Adjusting for axis of rotation not on coordinate axes