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Area, Volume and Perimeter

March 23, 2008

Consider the regions $\displaystyle \mathcal{R}_{1}\,\mathrm{and}\,\mathcal{R}_{2}$ enclosed by the curves $\displaystyle y=x^{3}+\frac{1}{2}$ and $\displaystyle x=-\left(y-\frac{1}{2}\right)^{2}$ . Set-up the integrals representing the following:

The perimeter of $\displaystyle \mathcal{R}_{2}$ .
The total area of the regions $\displaystyle \mathcal{R}_{1}$ and $\displaystyle \mathcal{R}_{2}$ .
The volume of the solid generated when $\displaystyle \mathcal{R}_{1}$ is rotated about the line $\displaystyle y=-\frac{1}{2}$ using the method of washers.
The volume of the solid generated when $\displaystyle \mathcal{R}_{2}$ is rotated about the line $x=-1$ using the method of cylindrical shells.

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5 Comments | Calculus | Tagged: arc length, area, cylindrical shells, volume, washers | Permalink
Posted by wMw

Largest Rectangle in Some Region in the First Quadrant

March 23, 2008

Find the area of the largest rectangle that can be inscribed in the region bounded by the curve $y=e^{-x}$ , the line $x=0$ and the positive $x$ -axis.