Evaluate $\int_C (2x^3-y^3)dx+(x^3+y^3)dy$, where $C$ is the unit circle.
Answer
By Green's Theorem we have \[\int_C(2x^3-y^3)dx+(x^3+y^3)dy=\iint_{D} \frac{\partial (x^3+y^3)}{\partial x}-\frac{\partial (2x^3-y^3)}{\partial y}dxdy\] \[ =\iint_{D} 3x^2+3y^2 dx dy=\int_{0}^{2 \pi} \int_0^{1}3r^2 r dr d\theta=2\pi \int_0^{1}3r^3 \] \[=2\pi(\frac{3}{4})=\frac{3\pi}{2}.\]
![Erdos](https://matchmaticians.com/storage/user/100028/thumb/matchmaticians-3empnt-file-5-avatar-512.jpg)
4.7K
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
- 2328 views
- $4.00
Related Questions
- Find the absolute extrema of $f(x,y) = x^2 - xy + y^2$ on $|x| + |y| \leq 1$.
- Sinusodial graph help (electrical)
- Determine where the following function is discontinuous
- Prove that $\int_{-\infty}^{\infty}\frac{\cos ax}{x^4+1}dx=\frac{\pi}{2}e^{-\frac{a}{\sqrt{2}}}(\cos \frac{a}{\sqrt{2}}+\sin \frac{a}{\sqrt{2}} )$
- Convergence of integrals
- What is this question asking and how do you solve it?
- A constrained variational problem
- Convergence of $\int_{1}^{\infty} e^{\sin(x)}\cdot\frac{\sin(x)}{x^2} $