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Find the volume generated when the region bounded by $y=x^3$ and the $x$-axis between $x=2$ and $x=7$ is rotated through $360^{\circ}$ about the $x$-axis.

Here is my attempt is this correct? \begin{align*} y &= x^3 \, , \; 2\le x\le7 \\ V &= \int_a^b \pi y^2 \, dx \\ &= \int_2^7\pi(x^3)^2 \, dx \\ &= \pi\int_2^7x^6 \, dx \\ &= \pi \left[ \frac{x^7}7 \right]_2^7 \\ &= \pi \left( 117649-\frac{128}{7} \right) \\ &= \frac{823415\pi}{7} \end{align*}

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    $\begingroup$ Looks good to me. $\endgroup$ – Arthur Feb 3 '17 at 7:43
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    $\begingroup$ Yup it's correct: verification $\endgroup$ – Hungry Blue Dev Feb 3 '17 at 8:09
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    $\begingroup$ This is even better. $\endgroup$ – Hungry Blue Dev Feb 3 '17 at 8:13

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