Some double integration help/Green's Theorem

[wootluke]

I have a few here so bear with me.
Problem 1: Compute this . I keep getting 0 on this one but I'm not sure if it is correct.
Problem 2: Compute this integral from C, where the curve C is (x,y) = (t, t² ), 0 <= t <= 1. I got 1 for this one but once again I have no clue if I was on the right track.
Here's where it gets tough for me.
Problem 3: Part a: Compute this integral where the curve C is the unit circle at (0,0) with counterclockwise orientation. I got pi for this one.
Part b: Compute the this integral where the curve C is the UPPER unit circle centered at (0,0) with a counterclockwise orientation. For this one I got pi/2.
Any help would be greatly appreciated! As you can see I got answers for all of them but I am unsure if I approached them the right way!

 

[Deleted member 4993]

I have a few here so bear with me.
Problem 1: Compute this View attachment 2823. I keep getting 0 on this one but I'm not sure if it is correct.
Problem 2: Compute this integral View attachment 2824 from C,where the curve C is (x,y) = (t, t² ), 0 <= t <= 1. I got 1 for this one but once again I have no clue if I was on the right track.
Here's where it gets tough for me.
Problem 3: Part a: Compute this integral View attachment 2825where the curve C is the unit circle at (0,0) with counterclockwise orientation. I got pi for this one.
Part b: Compute the this integral View attachment 2826 where the curve C is the UPPER unit circle centered at (0,0) with a counterclockwise orientation. For this one I got pi/2.
Any help would be greatly appreciated! As you can see I got answers for all of them but I am unsure if I approached them the right way!

Your problem "images" got cut-off. Correct those and re-post.

 

[Jason76]

Post Edited 5/10/2013

\(\displaystyle \int^{1}_{-1} \int^{1 - x^{2}}_0 (1 + x^{2} + y^{2})^{9}dy dx\)

\(\displaystyle \int^{1}_{-1} [\int^{1 - x^{2}}_0 (1 + x^{2} + y^{2})^{9}dy]dx\)

First, the goal is to integrate the inner terms with respect to y (partial integration). Next the integrated answer is evaluated with the inner definite integral values. Now this result is re-integrated with respect to x (but the y values have disappeared), and then evaluated (with the outer definite integral values) to get the final answer.

But the 9 exponent might demand substitution, so more work may be needed.

 

[Deleted member 4993]

tex]\int^1_-1 \int^{1 - x^{2}}_0 (1 + x^{2} + y^{2})^{9})dx dy[/tex]

\(\displaystyle \int^1_-1 [\int^{1 - x^{2}}_0 (1 + x^{2} + y^{2})^{9})dx]dy\)

The goal is to integrate the inner terms with respect to x and the outer terms with respect to y (partial integration).

Jason,

This is an advanced problem - requires different method of attack.

 

[daon2]

i'd go with polar on this one...

 

[Jason76]

The 9 exponent might demand a need for substitution. Otherwise, it would be just a simple double integration problem.

 

[wootluke]

I've decided to add photos of my work so you guys can see my thought process and hopefully jump off from there.

 

[DrPhil]

Post Edited after Response to it

\(\displaystyle \int^{1}_{-1} \int^{1 - x^{2}}_0 (1 + x^{2} + y^{2})^{9}dx dy\)

\(\displaystyle \int^{1}_{-1} [\int^{1 - x^{2}}_0 (1 + x^{2} + y^{2})^{9}dx]dy\)

First, the goal is to integrate the inner terms with respect to x (partial integration). Next the integrated answer is evaluated with the inner definite integral values. Now this result is re-integrated with respect to y (but the x values have disappeared), and then evaluated (with the outer definite integral values) to get the final answer.

But the 9 exponent might demand substitution, so more work may be needed.

WRONG - you have switched to order of integration!!

You have to integrate dy first, because its limits involve x. The result of the integration is a function of x.

THEN you can integrate dx from x=-1 to x=1.

 

[Jason76]

WRONG - you have switched to order of integration!!

You have to integrate dy first, because its limits involve x. The result of the integration is a function of x.

THEN you can integrate dx from x=-1 to x=1.

It depends. It can be:

\(\displaystyle \int^d_c [\int^b_a f(x,y) dx] dy\) (slab parallel to xz plane) (iterated definite integral)

or

\(\displaystyle \int^b_a [\int^d_c f(x,y) dy] dx\) (slab parallel to yz plane)

depending on what the problem requires. That's what my book says.

Looking at this problem again, it needs, \(\displaystyle \int^b_a [\int^d_c f(x,y) dy] dx\), as you said.

Actually, in regards to the problem, (due to the 9 exponent) you would substitution and the power rule before evaluation.

 

[DrPhil]

It depends. It can be:

\(\displaystyle \int^d_c [\int^b_a f(x,y) dx] dy\) (slab parallel to xz plane) (iterated definite integral)

or

\(\displaystyle \int^b_a [\int^d_c f(x,y) dy] dx\) (slab parallel to yz plane)

depending on what the problem requires. That's what my book says.

Looking at this problem again, it needs, \(\displaystyle \int^b_a [\int^d_c f(x,y) dy] dx\), as you said.

Actually, in regards to the problem, (due to the 9 exponent) you would substitution and the power rule before evaluation.

The limits are not the same if you change order of integration. The upper limit of the y-integration is y=1-x^2, a function of x, so this can not be taken outside the x-integration.

The area of integration is bounded by the x-axis and by the parabola y=1-x^2. Changing the order becomes

\(\displaystyle \displaystyle \int_0^1 \left[ \int_{-\sqrt{1-y}}^{\sqrt{1-y}} (1 + x^2 + y^2)^9\ dx \right] dy \)

EDIT: looking at your notes for question 2, you have done exactly the same thing!!
.........EXCEPT you have a additional factor of x that makes the function odd with respect to x,
......................so that the x-integral is zero.

Does that factor belong there in the original question? That would indeed make the answer zero.

\(\displaystyle \displaystyle \int_{-a}^a x\ f(x^2)\ dx = 0 \)