Question: Find the volume common to the cylinder $x^2+y^2=a^2$ and $x^2+z^2=a^2$ .

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$\text{Volume} = 8 \int\int\int dx \hspace{0.1cm}dx \hspace{0.1cm}dy \hspace{0.1cm}dz\\ \hspace{0.1cm}= 8 \int \int \int_{z=0}^{\sqrt{a^2-x^2}}dx \hspace{0.1cm}dy \hspace{0.1cm}dz\\ \hspace{0.1cm} = 8 \int\int(\sqrt{a^2 – x^2})dx \hspace{0.1cm}dy$

Now in the XY plane we have a circle $x^2 +y^2 = a^2$, y varies from 0 to $\sqrt{a^2-x^2}$ and x varies from 0 to a

$V = 8 \int_0^a\int_0^{\sqrt{a^2-x^2}}\sqrt{a^2}-x^2 dx \hspace{0.1cm}dy\\ \hspace{0.1cm} = 8 \int_0^a\Big[\sqrt{a^2 – x^2} . y\Big]_0^{\sqrt{a^2-x^2}}dx= 8 \int_0^a(a^2-x^2)dx\\ \hspace{0.1cm} = 8 \Big[a^2x - \frac{x^3}{3} \Big]_0^a = 8. \frac{2a^3}{3} = \frac{16a^3}{3}$

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