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:

  1. The perimeter of \displaystyle \mathcal{R}_{2}.
  2. The total area of the regions \displaystyle \mathcal{R}_{1} and \displaystyle \mathcal{R}_{2}.
  3. 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.
  4. 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: , , , , | Permalink
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Arc Length of an Astroid

March 3, 2008

Sketch the curve with equation x^{2/3}+y^{2/3}=1 and use symmetry to find its length.

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2 Comments | Calculus | Tagged: , , , | Permalink
Posted by wMw