# How to figure out the bounds of double integration?

I have a double integration problem that I do not understand how the bounds are what they are. The question is:

Find the volume of the given solid under the surface $$z = 5xy$$ and above the triangle with vertices $$(1, 1), (4, 1),$$ and $$(1, 2)$$.

Why are the bounds for this problem $$(1 \leq y \leq 2$$ and $$1 \leq x \leq 7-3y)$$ instead of $$(7-3y \leq x \leq y-1$$ and $$1 \leq y \leq 4)$$?

• The largest value $y$ takes is $2.$ Why would you say $1 \leq y \leq 4?$ – saulspatz Feb 28 '19 at 2:35
• If you look first to $y$-axis variation ($1 \leq y \leq 2$), imagine an arrow indicating the $x$-axis variation. When you think in the total variation, you can imagine an expanding point. When $y$ begins at $1$, the varitation of $x$ begins at $x = 1$ and ends at $x = 4$. When $y$ grows, the inicial variation of $x$ is still $x = 1$, but the final depends on $y$. – Corrêa Feb 28 '19 at 2:45

One way to find the bounds is to use the vertices of the triangle to help you. The hypotenuse of the triangle is defined by the segment connecting $$(4,1)$$ and $$(1,2)$$ if you find the equation of the line that correspond to this segment you can get a better idea of what the bounds are. Since you have a simple shape (a right triangle), you can set either the x or y variables to be the independent variable. Let's take the y variable as the independent variable.
Start by finding the slope of the segment $$\left(\frac{1-2}{4-1}\right) = \frac{-1}{3}$$ Then, you can use the point $$(1,1)$$ and the slope intercept form formula to find the equation of the line, which results in this: $$y = \left(\frac{-1}{3}\right)x + \left(\frac{2}{3}\right)$$ Solving for x gives: $$x = 2-3y$$ This equation is dependent on y and determines how far you can go on the x direction. The upper bound is the equation and the lower bound is the smallest x value in your set of points. Thus, $$(1 \leq x \leq 2-3y)$$ Since y was made to be the independent variable, all you need to do is find the smallest and largest y values in your set of points. So, using $$(1,1)$$ and $$(1,2)$$ we can tell that the y bounds are $$1 \leq y \leq 2$$