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posted by  Headache... on 11/15/2009 4:06:29 PM  |  status: Closed  |  Earned Karma: 50

Double integration Lifesaver

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Vector Calculus N/A N/A N/A 11/15/2009 at 6:00:00 PM
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Use double integration to find the area of the region enclosed by the curves and .
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posted by anonymousxyz on 11/15/2009 4:59:13 PM  |  status: Live
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\begin{align*} V &= \int_{-1}^3 \int_{y^2}^{2y+3} \;\text{d}x\,\text{d}y \\\\&= \int_{-1}^3 x\;\bigg|_{y^2}^{2y+3} \;\text{d}y \\\\&= \int_{-1}^3 2y + 3 - y^2 \;\text{d}y \end{align*}


     \begin{align*} \\\\&= y^2 + 3y - \frac{1}{3}y^3 \;\bigg|_{-1}^3 \\\\&= \left( 9 + 9 - 9 \right) - \left( 1 - 3 + \frac{1}{3} \right) \\\\&= 11 - \frac{1}{3} \\\\&= \frac{33}{3} - \frac{1}{3} \\\\&= \frac{32}{3} \end{align*}
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