Calculate the line integral along $\int \mathbf { F } \cdot d \mathbf { r } \text { for } \mathbf { F } ( x , y ) = 3 y \mathbf { i } + 5 x ^ { 2 } \mathbf { j } \text { and } C \text { is any path }$ starting at the point $( 3,5 )$ and ending at $( 8,0 )$.
A) $\frac { 10,150 } { 3 }$
B) $- 25$
C) $- 50$
D) 120
E) $- \frac { 10,565 } { 6 }$

Question

Quizplus · Accepted Answer

The vector field $\mathbf{F}(x, y) = 3y\mathbf{i} + 5x^2\mathbf{j}$ is conservative because it can be derived from a potential function $\phi(x, y)$ such that $\nabla \phi = \mathbf{F}$. To find the potential function, integrate $3y$ with respect to $x$ and $5x^2$ with respect to $y$, yielding $\phi(x, y) = 3xy + \frac{5}{3}x^3 + C$. The line integral of a conservative field over a path from point A to B is simply $\phi(B) - \phi(A)$. Substituting the given points, $\phi(8, 0) - \phi(3, 5) = (3 \cdot 8 \cdot 0 + \frac{5}{3} \cdot 8^3) - (3 \cdot 3 \cdot 5 + \frac{5}{3} \cdot 3^3) = 0 + \frac{5}{3} \cdot 512 - (45 + \frac{5}{3} \cdot 27) = \frac{2560}{3} - \frac{45 \cdot 3 + 45}{3} = \frac{2560 - 180}{3} = -\frac{1065}{6}$.

Calculate the Line Integral Along Starting at the Point (3,5)( 3,5 )(3,5)

Calculate the Line Integral Along Starting at the Point $( 3,5 )$