# If a region is bounded by y = sqrt(x) + 3, y=5, and the y=axis and it is revolved around the y = 7, how do you find the volume?

Nov 18, 2016

The volume is $= 120 \pi$

#### Explanation:

The volume of a small slice,

$\mathrm{dV} = \pi {\left(7 - \left(\sqrt{x} + 3\right)\right)}^{2} \mathrm{dx}$

Therefore,

$V = {\int}_{0}^{4} \pi {\left(7 - \left(\sqrt{x} + 3\right)\right)}^{2} \mathrm{dx}$

$V = \pi {\int}_{0}^{4} \left(49 - \left(x + 6 \sqrt{x} + 9\right)\right) \mathrm{dx}$

$= \pi {\int}_{0}^{4} \left(49 - x - 6 \sqrt{x} - 9\right) \mathrm{dx}$

$= \pi {\left[40 x - {x}^{2} / 2 - 6 {x}^{\frac{3}{2}} / \left(\frac{3}{2}\right)\right]}_{0}^{4}$

$= \pi \left(160 - 8 - 32\right) = 120 \pi$

graph{(y-(sqrtx+3))(y-5)(y-7)=0 [-5.335, 10.465, 2.226, 10.126]}