I am asked the following problem:
Evaluate the integral $$\iiint_{V} (2x+1) \, dx\,dy\,dz$$ in spherical coordinates given that $V$ is the region defined by: $$V = \lbrace x^2+y^2 \leq 9 \mid 0 \leq z\leq y, \ \ y \geq 0\rbrace.$$
What I have so far is
$$\int_{0}^{\pi} \int_{\pi/4}^{\pi/2} \int_{0}^{3 \csc(\theta) \csc(\phi)} (2\rho\cos(\theta)\sin(\phi)+1) \cdot \rho^2 \sin(\phi) \, d \rho\, d \phi\, d \theta$$
which makes perfect sense to me. What am I getting wrong?
Textbook's final answer: $18$
Thank you.