Compute $\oint_C y^2dx+3xydy $ where where $C$ is the counter clickwise oriented boundary of upper-half unit disk
Answer
Let $D$ be the upper-half unit disk. Using Green's theorem we get
\[\oint_C y^2dx+3xydy =\iint_D \frac{\partial}{\partial x}(3xy)-\frac{\partial}{\partial y}(y^2)dxdy\]
\[=\iint _D y dx dy=\int_{-1}^{1}\int_{0}^{\sqrt{1-x^2}}ydydx\]
\[=\int_{-1}^{1} \frac{y^2}{2}\Big|_0^{\sqrt{1-x^2}}dx =\int_{-1}^{1} \frac{1-x^2}{2}\]
\[=\frac{1}{2}(x-\frac{x^3}{3})\Big|_{-1}^{1}=\frac{1}{2}\Big(\frac{2}{3}- (-\frac{2}{3})\Big)=\frac{2}{3}.\]
Erdos
4.8K
-
Leave a comment if you need any clarifications.
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 3817 views
- $4.00
Related Questions
- Find all values of x... (Infinite Sums)
- In what direction the function $f(x,y)=e^{x-y}+\sin (x+y^2)$ grows fastest at point $(0,0)$?
- Measure Theory and the Hahn Decomposition Theorem
- Use Rouche’s Theorem to show that all roots of $z ^6 + (1 + i)z + 1 = 0$ lines inside the annulus $ \frac{1}{2} \leq |z| \leq \frac{5}{4}$
- Calc 3 Question
- Compute the surface integral $ \int_S (∇ × F) \cdot dS $ for $F = (x − y, x + y, ze^{xy})$ on the given surface
- Use Stokes's Theorem to evaluate $\iint_S ( ∇ × F ) ⋅ d S$ on the given surface
- Fermat's method of calculus
