- #1

tony873004

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## Homework Statement

Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem. [tex]\oint_C {xy^2 \,dx\, + \,x^3 \,dy} [/tex]. C is the rectangle with vertices (0,0), (2,0), (2,3) and (0,3).

## The Attempt at a Solution

I get the correct answer using Green's theorem. But I don't know how to do the "directly" method. I came up with 4 parametric equations, one for each line of the rectangle. Let's look at the first one for the line (0,0) - (2,0): x=t, y=0, t goes between 0 and 2.

[tex]\oint_C {xy^2 \,dx\, + \,x^3 \,dy} = \int\limits_0^2 {t\left( 0 \right)^2 \,\, + \,t^3 \,dt} = \int\limits_0^2 {t^3 \,dt} = \left[ {\frac{{t^4 }}{4}} \right]_0^2 = \left( {\frac{{2^4 }}{4}} \right) - \left( {\frac{{0^4 }}{4}} \right) = \frac{{16}}{4} = 4[/tex].

I'm pretty sure I did this step wrong. How do I go from the C integral to the 2-0 integral?

Thanks