# Evaluate the integral to use Fubini's Theorem

• Evaluate the integral $$\int_{E} f$$ where $$f(x, y, z)=x y+z$$ and $$E$$ is the tetrahedron in $$\mathbb{R}^{3}$$ with vertices $$(0,0,0),(1,0,0),(0,1,0)$$ and $$(0,0,1).$$ Use Fubini's theorem

My Attempt.

Note that $$\{(x,y,z): 0\leq x+y+z\leq 1\}$$, then $$\{0\leq x\leq 1, 0\leq y\leq 1-x, 0\leq z<1-x-y\}$$.

May you check my attemp, and may you write boundeds of integrals?

• Would you like to attempt to write the integral as written? I think your inequalities work. – Ninad Munshi Dec 23 '19 at 9:21

What you have written is correct. The integral is $$\int_0^{1}\int_0^{1-x}\int_0^{1-x-y} (xy+z) dzdydx$$.