Find volume where the region is bounded by $z=1-4(x^2+y^2)$ and the xy plane?

Question

Find volume where the region is bounded by $z=1-4(x^2+y^2)$ and the xy plane

I used cylindrical coordinate system to setup the integral $1-4(x^2+y^2)\geq z \geq 0$ $\Rightarrow$ $1-4r^2\geq z \geq 0$ and projection on xy plane is $x^2 + y^2 = 1/4$ or $r=1/2$

$\int_{0}^{2 \pi} \int_{0}^{1/2} \int_{0}^{1-4r^2} rdzdrd{\theta}$

on solving this I am getting $\frac {\pi}{8}$ as answer but the book lists $\frac{7\pi}{2}$ as answer $\cdots$ what am I doing wrong?

Answer 1

I think your answer is correct. Probably the book is wrong.