# Double Integral Set Up

The question was stated as follows,

Evaluate the following double integral;

$$\iint_R x^3y dA$$

where R is interior of triangle with vertices (0,0), (1,0), & (1,1) .

I thought for these types of double integrals I could have the limits in either of the two formats below;

$$\int_0^1\int_0^y x^3y dxdy$$ [1]

OR

$$\int_0^1\int_0^x x^3y dydx$$ [2]

However the answer sheet states that the answer for only the latter integral is correct.

How can I set up or visualize the problem in order to set it up in the correct way?

I did plot the points on a graph, and got my y=x "limit" from there, I just cannot understand why [2] is the correct integral, and how to go about making sure that in every double integral problem, I choose the correct integral set up.

All help is much appreciated :)

## 4 Answers

For the first one the bounds on x are wrong. The region is bounded by $y=x$ on the left and on the right by $x=1$. The bound on the left is a lower bound, so it should be

$$\int_0^1 \int_y^1 x^3ydxdy$$

• That makes sense! Thank you. – mnmakrets Jul 24 '15 at 14:10

You can draw the triangle.

Call the region inside of it $\Omega$.

Looking at lines of constant $y$, we'll have that $$\Omega = \{ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x \leq 1, y \leq x \leq 1 \}$$

Draw the triangle. Now think of it this way: if I fix $x$, what will be the extremes of integration for $y$? You'll get something that depends on $x$ and in particular $y$ goes from $0$ to $x$. Then integrate over the values taken on by $x$, and yo get the second form.

Alternatively, you can fix $y$ and notice that $x$ then varies from $y$ to $1$; then integrate over $y$ to get the correct form

$$\int_{0}^1 \int_y^1 x^3y dy dx$$

Usually you draw the area you want to integrate on, and draw the horizontal and vertical lines that corresponds to fixing $x$ or $y$. This helps visualize the problem and find the extremes of integration with ease

According fubini's theorem, you can either integrate parallel to the x-axis or the y-axis and hence there are two possibilities here:

$$\int_0^1 \int_0^x x^3y \ dy \ dx$$

or

$$\int_0^1 \int_y^1 x^3y \ dx \ dy$$

The best way to get to grips with these sort of questions is to practise more. For more, visit the following links: