double integral exponential region

Question

Hi I have to solve the following double integral.

$$\iint _{R} e^{x+3y}dA\: \:and \: \: R=\left \{ y=1, y=2, y=x, y=5-x \right \} $$

this is my approach:

$$\iint _{R} e^{x+3y}dA = \int_{1}^{2} \int_{1}^{x}e^{x+3y}dydx+\int_{2}^{3}\int_{1}^{2}e^{x+3y}dydx+\int_{3}^{4}\int_{1}^{5-x}e^{x+3y}dydx $$

Is this work correct? am I going in the right direction?

There should be inequalities rather than equalities in the definition of $R$. Otherwise, you're doing right! — Etienne, Jun 12 '14 at 16:43
I think I forgot to mention that the problem asks to calculate the volume of the region enclosed by the following lines. — Chico_Terry, Jun 12 '14 at 20:42

André Nicolas · Accepted Answer · 2014-06-12T16:37:28.133

3

The setup is correct. It is clear that you started the right way, by drawing a careful picture. And if you choose to integrate first with respect to $x$, you do have to consider three separate integrals.

But from that picture, one can see that it is much easier to integrate first with respect to $x$. So $x$ goes from $y$ to $5-y$, and then $y$ goes from $1$ to $2$. One calculation instead of three!

edited Jun 12 '14 at 16:37

answered Jun 12 '14 at 16:07

André Nicolas

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double integral exponential region

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