Study guide and
4 practice problems
on:
Ways of computing a line integral
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There are several ways to compute a line integral $\int_C \mathbf{F}(x,y) \cdot d\mathbf{r}$:
Direct parameterization
Fundamental theorem of line integrals
Green's theorem
Direct parameterization is the fall-back method. It is convenient when $C$ has a parameterization that simplifies $\mathbf{F}(x,y)$.
The fundamental theorem is usually the easiest to use, but it requires that $\mathbf{F}(x,y)$ is conservative.
Green's theorem only applies when $C$ is closed and $\mathbf{F}$ is well-behaved inside $C$.
Related topics
Line integrals
(8 problems)
Multivariable calculus
(147 problems)
Practice problems
Find $\int_C \bigl( \frac{1}{x+y} \mathbf{i} + \frac{1}{x+y} \mathbf{j} \bigr) \cdot d\mathbf{r}$, where $C$ is the segment of the unit circle going counterclockwise from $(1,0)$ to $(0,1)$.
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
Let $\mathbf{F}(x,y) = \langle 2, 3 \rangle$. Suppose $C$ is a curve connecting $(0,0)$ to $(1,1)$. Does the value of $\int_C \mathbf{F}\cdot d\mathbf{r}$ depend on the shape of the curve $C$? If not, find the value of the integral.
Solution
Let $C$ be a circle of radius $a$ centered at the origin, traversed counterclockwise. For what nonzero value of $a$ is $\oint_C \bigl(-y + \frac{1}{3} y^3 + x^2 y \bigr) dx = 0$?
Solution