Question

if

int_{c} mathbf{f} cdot d mathbf{r}=int_{c}(-3 x+2 y^{3}) d x+(x^{3} y) d y, then what is the field mathbf{f}?

mathbf{f}=

Explanation:

Step1: Recall the line integral formula

The line integral \(\int_{C}\mathbf{F}\cdot d\mathbf{r}=\int_{C}Pdx + Qdy\), where \(\mathbf{F}=\langle P,Q
angle\)

Step2: Identify \(P\) and \(Q\)

Given \(\int_{C}\mathbf{F}\cdot d\mathbf{r}=\int_{C}(- 3x + 2y^{3})dx+(x^{3}y)dy\), so \(P=-3x + 2y^{3}\) and \(Q = x^{3}y\)

Step3: Write the vector field \(\mathbf{F}\)

\(\mathbf{F}=\langle -3x + 2y^{3},x^{3}y
angle\)

Answer:

\(\mathbf{F}=\langle -3x + 2y^{3},x^{3}y
angle\)