Two Variable Calculus Problems with Ellipse and Hyperbola Integrals

[Kate2010]

Homework Statement

I have several problems:

1. Calculate $$\int\$$ F . dr over C
F = (y²,-x²) and C is the portion of the ellipse x² + 4y² = 4 which lies in the positive quadrant.

2. Calculate $$\int$$ (3x² + 3y²)^1/2 ds where C is the part of the hyperbola x² - y² = 1 from (1,0) to (cosh2, sinh2).

Homework Equations

The Attempt at a Solution

1. I have worked this out and got -20/3 but I'm unsure if it can be negative? I'm not very confident with working them out so am unsure about my answer. I can post more working if this is incorrect.

2. I assume I need to use the parameters x=cosht, y=sinht and integrate from 0 to 2. However, I am confused as to what I multiply by after I have subbed in for x=cosht and y=sinht, before I integrate, if that makes sense.

 

[mighty2000]

For the first problem, your answer of -20/3 is correct. Since the integral is over a closed curve, the result can be negative.

For the second problem, you are correct in using the parameterization x=cosh(t), y=sinh(t). After substituting these values in the integral, you will need to multiply by the derivative of the parameterization, which is (cosh(t), sinh(t)). This will give you a ds term in the integral, which you can then integrate from 0 to 2. I hope this helps. Good luck with your calculations!