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 $1 \leq y \leq 4)$?

  • $\begingroup$ The largest value $y$ takes is $2.$ Why would you say $1 \leq y \leq 4?$ $\endgroup$ – saulspatz Feb 28 '19 at 2:35
  • $\begingroup$ 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 $y$. $\endgroup$ – 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 $$1 \leq y \leq 2$$

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