# Evaluate the double integral?

Math10

## Homework Statement

Evaluate the double integral (x+2y)dA, where R is the region in the first quadrant bounded by the circle x^2+y^2=9.

None.

## The Attempt at a Solution

I know how to evaluate the double integral but I just don't know how to find the limits of integration. I know that the radius of the circle is 3.

## Answers and Replies

Homework Helper
Gold Member
2021 Award
First, what are the limits for x?

Math10
I have no idea. Is it 0 to 3?

Homework Helper
Gold Member
2021 Award
It looks like you do have an idea. That's right.

For each x, what are the limits of y? You may need to draw the area.

Math10
From 0 to 3?

Mentor

## Homework Statement

Evaluate the double integral (x+2y)dA, where R is the region in the first quadrant bounded by the circle x^2+y^2=9.

None.

## The Attempt at a Solution

I know how to evaluate the double integral but I just don't know how to find the limits of integration. I know that the radius of the circle is 3.
How would you describe the region R as a set? IOW, R = {(x, y) | <inequality involving x> and <inequality involving y>}.

If you are tempted to write the inequalities as ##0 \le x \le 3, \text{and } 0 \le y \le 3##, resist that temptation! That region would be a square, which the one in this problem is not.

As I said in another of your threads, many times carrying out the integration is the easy part, requiring very little thinking. Figuring out the limits of integration is often the more difficult task. Your textbook should have a number of examples. Have you looked at them?

• Math10
Homework Helper
Gold Member
2021 Award
From 0 to 3?
That would be a square in the first quadrant.

Math10
So y=sqrt(9-x^2) and y= - sqrt(9-x^2)?

Homework Helper
Gold Member
2021 Award
So y=sqrt(9-x^2) and y= - sqrt(9-x^2)?
Why would y be negative?

Math10
So y=0 to sqrt(9-x^2)?

Mentor
So y=0 to sqrt(9-x^2)?
Yes. Now what are your limits for x?

• Math10
Math10
0 to 3?

Mentor
Yes. Now you need to figure the order in which you're going to integrate.

Math10
dy dx?

Mentor
dy dx?
You tell me ...

Math10
dy dx.

Mentor
OK. Now you're ready to set up the iterated integral and evaluate it...

• Math10
Math10
Thank you!

Mentor
You're welcome!