Study guide and
2 practice problems
on:
Computing line integrals directly
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To compute the line integral $\int_C \mathbf{F}\cdot d\mathbf{r}$ directly, we
Parameterize the curve $C$
Express $\mathbf{F} \cdot d\mathbf{r}$ in terms of the parameter
Evaluate the resulting one-dimensional integral
Related topics
Line integrals
(8 problems)
Multivariable calculus
(147 problems)
Practice problems
Directly compute $\int_C \mathbf{F} \cdot d\mathbf{r}$ where $\mathbf{F}(x,y) = x y \ \mathbf{i} + \mathbf{j}$ and $C$ is the curve connecting $(0,0)$ to $(1,1)$ along $y=x^2$.
Solution
Let $\mathbf{F}(x,y) = \frac{-y \ \mathbf{i} + x \ \mathbf{j}}{\sqrt{x^2 + y^2}}$. Compute $\int_C \mathbf{F} \cdot d\mathbf{r}$ where $C$ is the circle of radius $r$, centered at the origin and traversed counterclockwise.
Solution