Find the volume of the solid whose base is bounded by the circle $x^{2}+y^{2}=4$ with the indicated cross sections taken perpendicular to the $x$ -axis. (a) Squares (b) Equilateral triangles
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The volume is calculated as: \[ V = 2\int_{-2}^{2} (\sqrt{4-x^{2}})^{2} dx = 2\int_{-2}^{2} (4-x^{2}) dx \] \[ = 2\left[ 4x - \frac{x^{3}}{3} \right]_{-2}^{2} = 2\left( 8 - \frac{8}{3} \right) = \frac{64}{3} \] Show more…
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Using Cross Sections Find the volumes of the solids whose bases are bounded by the circle $x^{2}+y^{2}=4,$ with the indicated cross sections taken perpendicular to the $x$ -axis. $\begin{array}{ll}{\text { (a) Squares }} & {\text { (b) Equilateral triangles }}\end{array}$
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