Evaluate the line integral $\int _ { C } 3 x ^ { 2 } y ^ { 2 } d x + 2 x ^ { 3 } y d y$ , where $C$ is any path from $( 1,2 )$ to $( 2,1 )$ .
A) $2$
B) $4$
C)$6$
D)$8$
E) $- 2$
F) $- 4$
G) $- 6$
H)$- 8$

Question

Evaluate the line integral $\int _ { C } 3 x ^ { 2 } y ^ { 2 } d x + 2 x ^ { 3 } y d y$ , where $C$ is any path from $( 1,2 )$ to $( 2,1 )$ .
A) $2$ 
B) $4$
C)$6$
D)$8$
E) $- 2$
F) $- 4$
G) $- 6$ 
H)$- 8$

Quizplus · Accepted Answer

The given line integral can be evaluated directly or by checking if the vector field is conservative. In this case, it's more straightforward to use the fundamental theorem for line integrals, given that the vector field is conservative. The vector field here is $\mathbf{F} = 3x^2y^2\mathbf{i} + 2x^3y\mathbf{j}$. To check if it's conservative, we can see if the curl of $\mathbf{F}$ is zero. However, a quicker method for this problem is to notice that the integral depends only on the start and end points, not the path taken, suggesting we can find a potential function $\Phi(x, y)$ such that $\nabla \Phi = \mathbf{F}$.A potential function $\Phi(x, y)$ would satisfy $\frac{\partial \Phi}{\partial x} = 3x^2y^2$ and $\frac{\partial \Phi}{\partial y} = 2x^3y$. Integrating $\frac{\partial \Phi}{\partial x} = 3x^2y^2$ with respect to $x$ gives $\Phi = x^3y^2 + g(y)$, and integrating $\frac{\partial \Phi}{\partial y} = 2x^3y$ with respect to $y$ gives $\Phi = x^3y^2 + h(x)$. Since both must be true, $g(y)$ and $h(x)$ are constants with respect to their integrations, and we can ignore them for the purpose of evaluating the line integral from one point to another.Evaluating the potential function at the start and end points gives:$$\Phi(2, 1) - \Phi(1, 2) = (2^3 \cdot 1^2) - (1^3 \cdot 2^2) = 8 - 4 = 4$$Therefore, the value of the line integral is $4$, which corresponds to choice B.

Evaluate the Line Integral ∫C3x2y2dx+2x3ydy\int _ { C } 3 x ^ { 2 } y ^ { 2 } d x + 2 x ^ { 3 } y d y∫C​3x2y2dx+2x3ydy

Evaluate the Line Integral $\int _ { C } 3 x ^ { 2 } y ^ { 2 } d x + 2 x ^ { 3 } y d y$