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I need someone to check my work.

so I know this is a upside down parabola so I find the two x coordinates which are

\(\displaystyle 16 - y^4 = 0\)

\(\displaystyle y^4 = 16\)

\(\displaystyle y^2 = +- \sqrt{4}\)

\(\displaystyle y = +- 2\)

so I know

\(\displaystyle \int^2_{-2} 16 - y^4 dy\)

Take antiderivative

\(\displaystyle 16y - \frac{1}{5}y^5\) | -2 to 2

so \(\displaystyle (2) = 32 - \frac{32}{5} = \frac{160}{5}\)

then \(\displaystyle (-2) = -32 - (\frac{-32}{5}) = -32 + \frac{32}{5} = \frac{-160}{5} + \frac{32}{5} = \frac{-128}{5}\)

SO finally

\(\displaystyle

[\frac{160}{5}] - [\frac{-128}{5}] = \frac{288}{5}\)