How to find bounds when doing a double integral?

Mathematics Asked by Dr. Suess Official on December 5, 2020

$$iint e^{x+y},dA$$

domain defined by:$${xge 5:::,:yge 5::,:x+yle 16}$$

This is what I’m given and I’m asked to solve this.

I know the formula $ale xle b::,:g_1left(xright)le yle g_2left(xright)$.

But I’m not sure how I would find those components.

I got this so far:

$$a=5, b=16$$

And I’m not sure what $g_1(x)$ or the other g is.

How would I find that?

One Answer

Make a sketch of those limits. For $xge 5$, draw a vertical line at that value, and you know that your domain is to the right. For $yge 5$, draw a horizontal line at $y=5$, and your domain is above that line. Similarly, $x+y=16$ is a line that connects $(16,0)$ and $(0,16)$, and your domain is below that.

With this information, your domain is a triangle. Calculate the intersection points. You should get the vertices at $(5,5)$, $(5,11)$ and $(11,5)$. Then $x$ varies between $5$ and $11$, and $y$ varies between $5$ and $16-x$. Or, if you change the order of integration, $y$ varies between $5$ and $11$, and $x$ between $5$ and $16-x$.

Correct answer by Andrei on December 5, 2020

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