Computing volume of a region

Question

The problem asks to compute the volume of the region $$ V = \{(x,y,z): 0 \leq z \leq x^2 - y^2, \; 0 \leq x^2 + y^2 \leq 1 \} $$

I am stuck at this ridiculously simple problem. The volume turns out to be $0$, and I am unable to work through the problem the correct way. It reminds me of the problem where the integral of the sine function over the interval $[0, 2\pi]$ turns out to be $0$, which makes no sense at all if we are actually finding the area under the sine curve. I have a feeling the above computation of volume gives us $0$ for the same reason. How do I get around this problem and compute the volume correctly?

Answer 1

The volume is the portion of a cylinder that intersects with the region between the $xy$-plane and the hyperbolic paraboloid $z=x^2-y^2\,.$

• This volume can impossibly be zero.

Due to symmetry we can restrict to $x,y\ge 0$ (see picture) and multiply the result by four.

• Since the paraboloid intersects with the $xy$-plane at the line $x=y$ we must calculate \begin{align} V&=4\int_0^1\int_0^x\int_0^{x^2-y^2}\,dz\,dy\,dx=4\int_0^1\int_0^xx^2-y^2\,dy\,dx\\ &=4\int_0^1x^3-\frac{x^3}{3}\,dx=\frac{2}{3}\,. \end{align} When instead you miss my second bullet point you might naively calculate \begin{align} V&=4\int_0^1\int_0^{\color{red}{\sqrt{1-y^2}}}\int_0^{x^2-y^2}\,dz\,dx\,dy \end{align} which is indeed zero.