Thread: double integral over general region

1. double integral over general region

Using the maxima and minima of the function, produce upper and lower estimates of the integral
where is the circular disk: .
??? ???

how to do that?

2. Maybe...

$\displaystyle e^{5 \cdot (0)} \le e^{5 \cdot (x^{2}+y^{2})} \le e^{5 \cdot (9)}$

3. i try to enter those in the answer box, but turn out that it's not right.
I try integrate those and i got : $\displaystyle 2e^{45}\pi -2\pi$ which is not right as well.

4. Perhaps you should try those integrals again.

For the lower bound, I get $\displaystyle \frac{9\pi}{4}$.

5. $\displaystyle \frac{9\pi}{4}$ which is not correct either.

6. Whoops. I did only one quadrant.

The question states, "produce upper and lower estimates". $\displaystyle 9\pi$ IS a lower bound.

Did you do as I asked? Did you try those integrals again or did you just type in my answer?

7. I used polar coordinate to estimate. For the lower bound, I try to integrate
r from 0 to 3 and $\displaystyle \theta$ from 0 to $\displaystyle \pi$

the answer turn out to be $\displaystyle e^{45} \pi - \pi$ and it is not correct.

In fact, i do not quite understand by "upper and lower estimates of the integral".

8. 1) Why [0,pi]? Try [0,2pi]

2) Why are you trying to evaluate the integral directly? That is not what is asked.

3) My first response entirely exposes the concept of bounds. We replace the argument with something simpler and see where it leads.