Calculate Integral $\iiint_{K}\frac{1}{(z+1)}\,dx \,dy \,dz $

Question

I need to calculate this integral

$$\iiint_{K}\frac{1}{(z+1)}\,dx \,dy \,dz $$

with $K=x^{2}\le y\le z\le x$

I have trouble visualizing $K$ so I am unable to find the extremities of the integrals

Any help would be much appreciated

Answer 1

You have $x^2\leqslant x$ when $x\in[0,1]$ and only on that interval. For each $x\in[0,1]$, $y$ can take any value from $x^2$ to $x$. And, for each such $y$, $z$ can take any value from $y$ to $x$. Therefore, you are after the integral$$\int_0^1\int_{x^2}^x\int_y^x\frac1{z+1}\,\mathrm dz\,\mathrm dy\,\mathrm dx.$$Can you take it from here?