Evaluate the line integral of a parabola How can I evaluate :
$$\int_{C} y \;dx + x^2 \; dy$$
where $C$ is the parabola define by
$$y=4x-x^2 \quad \text{from } \; (4,0) \; \text{ to } \; (1,3).$$

Do I need to parameterize the parabola?

$$\because y = 4x-x^2 , \therefore \frac{dy}{dx} = 4-2x , \therefore dy = (4-2x)dx $$
Hence,
$$\int_{C} y \;dx + x^2 \; dy = \int_{4}^{1} 4x-x^2 dx + \int_{4}^{1} x^2 (4-2x)dx$$
$$\implies \int_4^1 4x-x^2+4x^2-2x^3$$
$$\implies \int_4^1 4x+3x^2-2x^3$$
$$\implies (2x^2+x^3-\frac{x^4}{2})_4^1$$
$$\implies (2+1-\frac{1}{2}) - (32+64-128)$$
$$\implies 35-\frac{1}{2}$$
$$\implies \frac{69}{2}$$
