# Multivariable Calculus Order of Integration Question

I have a triple integral $\int_{0}^{2} \int_{0}^{y^3} \int_{0}^{y^2} f(x,y,z) \ dz \ dx \ dy$. I have to find five different iterated integrals equivalent to this integral. I know that order of integration involves projecting the graph onto the $xy,\ xz$, and $yz$ planes but I am not really sure what to do beyond that.

EDIT: I came with the following solution: $$\int_{0}^{2} \int_{0}^{y^2} \int_{0}^{y^3} f(x,y,z) \ dx \ dz \ dy \\ \int_{0}^{8} \int_{\sqrt{x}}^{2} \int_{0}^{y^2} f(x,y,z) \ dz \ dy \ dx \\ \int_{0}^{4} \int_{\sqrt{z}}^{2} \int_{0}^{y^3} f(x,y,z) \ dx \ dy \ dz \\ \int_{0}^{4} \int_{0}^{8} \int_{2}^{\sqrt{x}} f(x,y,z) \ dy \ dx \ dz \\ \int_{0}^{8} \int_{0}^{4} \int_{\sqrt{z}}^{2} f(x,y,z) \ dy \ dz \ dx$$ Would this be correct?

• There are 6 ways to order 3 things. You already have one ordering in your integral $dz\ dx\ dy$. The other 5 come from changing the order of integration. The tricky part will be correctly changing the limits of integration. – John Habert Jan 30 '14 at 2:01

You have: $$\int_0^2 \left( \int_0^{y^3} \left(\int_0^{y^2} f(x,y,z) \ dz\right) \ dx \right) \ dy.$$
Look at the outside integral: $$\int_0^2 \cdots\cdots\cdots\,dy.$$ The variable $y$ goes from $0$ to $2$. Inside this integral you have $$\int_0^{y^3} \cdots\cdots\cdots\,dx.$$ That means that, for each fixed value of $y$ between $0$ and $2$, the variable $x$ goes from $0$ to $y^3$. So $x$, being between $0$ and $y^3$, could be anywhere between $0$ and $2^3=8$. But $x$ cannot be more than $y^3$, or in other words $y^3$ must be more than $x$, or equivalently $y$ must be more than $\sqrt{x}$. Thus one can write $$\int_0^8\cdots\cdots\cdots\,dx,$$ and then say that for any value of $x$ between $0$ and $8$, the variable $y$ goes from $\sqrt{x}$ to $2$. Thus we have $$\int_0^8\left( \int_{\sqrt{x}}^2 \cdots\cdots\cdots\,dy \right)\,dx.$$
• I came with the following solution $\int_{0}^{2} \int_{0}^{y^2} \int_{0}^{y^3} f(x,y,z) \ dx \ dz \ dy \\ \int_{0}^{8} \int_{\sqrt{x}}^{2} \int_{0}^{y^2} f(x,y,z) \ dz \ dy \ dx \\ \int_{0}^{4} \int_{\sqrt{z}}^{2} \int_{0}^{y^3} f(x,y,z) \ dx \ dy \ dz \\ \int_{0}^{4} \int_{0}^{8} \int_{2}^{\sqrt{x}} f(x,y,z) \ dy \ dx \ dz \\ \int_{0}^{8} \int_{0}^{4} \int_{\sqrt{z}}^{2} f(x,y,z) \ dy \ dz \ dx$. Would this be correct? – Jackie Jan 31 '14 at 0:31
• I think you need $\displaystyle \int_0^8 \int_0^4 \int_{\max\{\sqrt{z},\sqrt{x}\}}^2 f(x,y,z)\, dy\, dz \, dx$, and similarly for the other one with $\displaystyle \int\cdots\cdots \,dy$ as the innermost integral. – Michael Hardy Jan 31 '14 at 0:58
• The two constraints, that $z\le y^2$ and that $x\le y^3$, mean that $y\ge\text{both }\sqrt{z}\text{ and }\sqrt{x}$. If $y\ge\text{both of two things}$ then $y\ge\max\text{ of those two things}$. ${}\qquad{}$ – Michael Hardy Jan 31 '14 at 17:50