# 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?

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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