Find the volume enclosed by the planes $2 x + y = 3, \ z = y , \ y = x$, and $ z = 0$.

Question

Find the volume enclosed by the planes $2 x + y = 3, \ z = y , \ y = x$ , and $z = 0$.

Is it the integral:

$$ \int_{0}^{1}\int_{y}^{\frac{3-y}{2}}(y-0)dydx $$

Answer 1

The integral in your picture cannot be right since

$$\int _{0} ^{1} \int _{y} ^{\frac{3-y}{2}} (y-0)\, dy\,dx$$

means you are integrating with respect to $y$ first, then evaluating limits which are functions of $y$. After integrating with respect to a certain variable, you should no longer see that variable. On the other hand, the integral is close to correct, since

$$\int _{0} ^{1} \int _{y} ^{\frac{3-y}{2}} (y-0)\, dx\,dy$$

will give you the correct result.