# Evaluate a line integral using the fundamental theorem of line integrals

Consider the vector field $${\mathbf F}(x,y)=(e^x)(\sin y){\mathbf i}+(e^x)(\cos y){\mathbf j},$$ and the curve $C$ composed of the graph of $\sqrt{x}+\sqrt{y}=5$ followed by segment from $(25,0)$ to $(0,0)$. Evaluate the line integral $$\int_C {\mathbf F} \cdot d {\mathbf r}.$$ I noticed that ${\mathbf F}(x,y)$ is conservative, so I applied the Fundamental Theorem of Line Integral and got the potential function $$f(x,y)=(e^x)(\sin y),$$ matching my professor's answer key. I then did $f(0,0)-f(25,0)$ to get an answer of $0$, but in my professor's answer key he used $f(0,0)-f(0,25)$ to get an answer of $-\sin(25)$. Why is that?

## 1 Answer

\begin{align*} F(0,0)-F(0,25)&=e^{0}\sin(0)-e^0\sin(25) \\ &=1\cdot 0-1\cdot\sin(25)\\ &=-\sin(25) \end{align*}

Edit

The graph of $$\sqrt{x}+\sqrt{y}=5$$ is the portion of a parabola going from $(0,25)$ to $(25,0)$, which you must include as part of your integral. So the integral starts at $(0,25)$, traverses the parabola, then goes to $(0,0)$ along the $x-axis$. So the start point is $(0,25)$ and the end point is $(0,0)$.

• You are not addressing the OP's question. He understands that $f(0,0) - f(0,25) = - \sin(25)$. He does not understand why that is what he's supposed to do. – Mark Fantini Dec 7 '14 at 18:38
• Oh woops...ill edit. – illysial Dec 7 '14 at 18:38
• @Mark_Fantini I have made the appropriate edit – illysial Dec 7 '14 at 18:44
• I have reversed the downvote and upvoted. :) – Mark Fantini Dec 7 '14 at 18:45