(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Evaluate [tex]\displaystyle \int_C y^2dx + x^2dy[/tex] for the path C: the boundary of the region lying between the graphs of [tex]\displaystyle y=x[/tex] and [tex]\displaystyle y=\frac{x^2}{4}[/tex].

2. Relevant equations

The catch is that you can't use Green's Theorem.

3. The attempt at a solution

I think you can break C into two other curves, [tex]C_1:r(t)=t\textbf{i}+t\textbf{j}[/tex] for [tex]0 \leq t \leq 4[/tex] and [tex]C_2:r(t)=(8-t)\textbf{i}+\frac{(8-t)^2}{4}\textbf{j}[/tex] for [tex]4 \leq t \leq 8[/tex].

I believe my error is somewhere below:

[tex]

2\int_{C_1}t^2dt+\frac{-1}{2} \int_{C_2}(8-t)^3((8-t)+1)dt \Rightarrow 2\int_0^4t^2dt+\frac{-1}{2} \int_4^8(8-t)^3((8-t)+1)dt

[/tex]

I get that expression equal to [tex]\frac{-1376}{15}[/tex], which I know is incorrect, as I used Green's Theorem, and got [tex]\frac{32}{15}[/tex] which also coincides with the answer key I have.

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# Homework Help: Green's Theorem well, sort of.

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