To evaluate the line integral $\int_C 2y \, dx$ over the triangular path $C$ with vertices $(0,0)$, $(3,0)$, and $(5,4)$, we can approach this by breaking the path $C$ into segments and calculating the integral over each segment separately.

Step 1: Define the Segments

1. Segment 1: From $(0,0)$ to $(3,0)$.

   • Here, $y = 0$, so $2y = 0$, which makes this integral trivially zero for this segment.

2. Segment 2: From $(3,0)$ to $(5,4)$.

   • The equation of the line segment between these two points can be written as: $y = 2(x - 3)$
   • Solving for $y$ in terms of $x$ will allow us to express $2y$ in terms of $x$ and perform the integration with respect to $dx$.

3. Segment 3: From $(5,4)$ back to $(0,0)$.

   • The equation of the line from $(5,4)$ to $(0,0)$ is: $y = \frac{4}{5}x$

Now, let's calculate the integral over each segment:

Calculation

1. Integral over Segment 1: $\int_{(0,0) \to (3,0)} 2y \, dx = 0$

2. Integral over Segment 2: Substitute $y = 2(x - 3)$, then $2y = 4(x - 3)$. $\int_{3}^{5} 4(x - 3) \, dx = \int_{3}^{5} (4x - 12) \, dx$ Evaluating this integral: $= \left[ 2x^2 - 12x \right]_{3}^{5} = \left( 2(5)^2 - 12(5) \right) - \left( 2(3)^2 - 12(3) \right)$ Simplifying each term: $= (50 - 60) - (18 - 36) = -10 + 18 = 8$

3. Integral over Segment 3: Substitute $y = \frac{4}{5}x$, so $2y = \frac{8}{5}x$. $\int_{5}^{0} \frac{8}{5}x \, dx = -\int_{0}^{5} \frac{8}{5}x \, dx$ Evaluating this integral: $= -\frac{8}{5} \int_{0}^{5} x \, dx = -\frac{8}{5} \cdot \frac{x^2}{2} \Big|_{0}^{5}$ $= -\frac{8}{5} \cdot \frac{25}{2} = -20$

Final Answer

Adding up the integrals from each segment: $0 + 8 - 20 = -12$

So, the result of the line integral is: $\boxed{-12}$

Would you like further details on any specific part of this calculation?

Here are some related questions that expand on this topic:

1. How do you find the equation of a line segment between two points?
2. What is the general method for evaluating line integrals?
3. How do orientation and parametrization affect line integrals?
4. Can you use Green's Theorem to simplify certain line integrals?
5. How do line integrals apply in physics and engineering?

Tip: Always check if Green's Theorem applies to simplify line integrals over closed curves.