Question

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

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}$$. The line is y = x$, or equivalently x = y$. Find the points of intersection by setting $$\frac{$$y^{2}$$}{4} = y$$. Solve the equation $$\frac{$$y^{2}$$}{4} = y$$ to find the intersection points. Multiply both sides by 4 to get y$$^{2}$$ = 4y$$, then rearrange to $$y^{2} - 4y = 0$, and factor as y(y - 4) = 0$. So, y = 0$ or y = 4$ are the limits of integration. Since the solid is generated by revolving the region about the y-axis, use the method of cylindrical shells. The formula for the volume is V$$ = \int_{a}^{b} 2\pi (	ext{radius})(	ext{height}) \, dy$$, where the radius is the distance from the y-axis and the height is the horizontal distance between the curves. Determine the radius and height for the shell at a given y$. The radius is the distance from the y-axis, which is x$. The height is the difference between the rightmost and leftmost x$ values for the region at that y$. Here, the right boundary is x = y$ (from the line) and the left boundary is x$$ = \frac{$$y^{2}$$}{4}$$ (from the parabola). So, height = y$$ - \frac{$$y^{2}$$}{4}$$. Set up the integral for the volume: V$$ = \int_{0}^{4} 2\pi y \left(y - \frac{$$y^{2}$$}{4}ight) dy$$. This integral represents the volume of the solid generated by revolving the region about the y-axis.

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

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Concetti chiave

Understanding the Region Bounded by Curves

Method of Cylindrical Shells for Volume

Setting Up and Evaluating Definite Integrals