# Evaluating Integrals on Ellipse: C and C

• teleport
In summary: No, I didn't mean to make a line connecting the two ellipses. I meant to make a cut between them using the x-axis. Then I would integrate around one ellipse and up that cut to the other ellipse. After that, I would integrate around the curve that resulted from that cut. But since the curve doesn't have (0,0) in its interior, the integral around the two ellipses would be 0.
teleport

## Homework Statement

(i) Evaluate

$$\int_C \dfrac{-ydx + xdy}{9x^2 + 16y^2}$$

when C is the ellipse

$$\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$$

(ii) Use the ans to (i) to evaluate the integral along C' = ellipse:

$$\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$$

## The Attempt at a Solution

I have done (i) but have no clue about (ii). great thnx for any help.

Okay, what did you get for (i)? I can think of a number of ways that might help you answer (ii) but since you have shown no work at all I don't know which way would be appropriate for you. Do you know Green's Theorem?

got pi/6 for i. didn't use green's for that. the other way is easier. for ii u can't use green's since it would either be too complicated or impossible to integrate. yes i know green's thm. sorry for not showing any work. i just don't know what to do for ii.

That's not at all what I get for (i). And the problem with (ii) is I don't know what theorems or methods you have available. I do notice that the integrand is defined everywhere except at (0,0) which is inside both ellipses.

checked it and still got the same. is it allowed to say 9x^2 + 16y^2 = 144 on that integral? that's something I'm using.

for ii, if u could mention some of the methods u have in mind, i might recognize it as something given in class. thnx

My apologies. I just screwed up (1) by copying part of the problem incorrectly. Yes, $\pi/6$ is the correct answer.

My point about the integrand not being defined at (0,0) was that you make a cut from the outer ellipse to the inner, integrate around one ellipse, then up that cut to the other and back. Then you are integrating around a curve that does NOT have (0,0) in its interior. The integral around that path will be 0, showing that the integral around the two ellipses is the same.

and why is it 0? what does the integrand not being defined at (0,0) has to do with that? by a cut do u mean to make a line connecting the 2 ellipses? could u explain more please?

## 1. What is the equation for an ellipse?

The equation for an ellipse is x2/a2 + y2/b2 = 1, where a and b represent the lengths of the major and minor axes, respectively.

## 2. How do you evaluate integrals on an ellipse?

To evaluate integrals on an ellipse, we first need to parameterize the ellipse using trigonometric functions. Then, we can use substitution or trigonometric identities to simplify the integral. Finally, we use integration techniques such as u-substitution or integration by parts to solve the integral.

## 3. What is the center of an ellipse?

The center of an ellipse is the point where the major and minor axes intersect. It is also the midpoint of the foci of the ellipse.

## 4. Can you use the same techniques to evaluate integrals on any type of ellipse?

Yes, the same techniques can be used to evaluate integrals on any type of ellipse, as long as it is a standard ellipse with its center at the origin.

## 5. Are there any special cases when evaluating integrals on an ellipse?

Yes, there are certain special cases when evaluating integrals on an ellipse. For example, if the ellipse is a circle, the integrals can be simplified using polar coordinates. Also, if the ellipse is rotated, we may need to use a different parameterization or apply a rotation transformation to the integral.

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