Find the volume of the solid under the plane $x + 2y - z = 0$ and above the region bounded by $y=x$ and $y = x^4$. Find the volume of the solid under the plane $x + 2y - z = 0$ and above the region bounded by $y = x$ and $y = x^4$.
$$
\int_0^1\int_{x^4}^x{x+2ydydx}\\
\int_0^1{x^2-x^8dx}\\
\frac{1}{3}-\frac{1}{9} = \frac{2}{9}
$$
Did I make a misstep? The answer book says I am incorrect.

$\int_0^1\int_{x^4}^x{(x+2y)\mathrm{d}y\mathrm{d}x}$ You have set correctly the integral but you did not integrate correctly.
Be careful in $\int_{x^4}^x{(x+2y)}\mathrm{d}y$ you integrate in respect of $y$ so you should find
$$\int_{x^4}^x{(x+2y)}\mathrm{d}y=\left | xy+y^2 \right |_{x^4}^{x}=x^2+x^2-(x^5+x^{8}) $$
And finally you get $\int_0^1 2x^2-x^5-x^{8}\mathrm{d}y=\frac{2}{3}-\frac{1}{6}-\frac{1}{9}=\frac{7}{18}$
You could verify your results here
