Volume of Parabolic Cylinder in 1st Octant: 710/3

[carl123]

Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 25 − x² and the plane y = 2.

I already solved it and got 710/3 as my answer, I just wanted to make sure its the right answer

 

[Prove It]

Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 25 − x² and the plane y = 2.

I already solved it and got 710/3 as my answer, I just wanted to make sure its the right answer

No, it's not the right answer, I get 500/3.

If you draw out the region you should see a prism, with its cross sections along the y-axis bounded by $\displaystyle \begin{align*} 0 \leq y \leq 2 \end{align*}$. If you look at each cross section, it is bounded vertically by $\displaystyle \begin{align*} 0 \leq z \leq 25 - x^2 \end{align*}$ and bounded horizontally by $\displaystyle \begin{align*} 0 \leq x \leq 5 \end{align*}$. So that means that the volume is calculated using

$\displaystyle \begin{align*} V = \int_0^2{\int_0^5{\int_0^{25 - x^2}{1\,\mathrm{d}z}\,\mathrm{d}x}\,\mathrm{d}y} \end{align*}$