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
4.8K
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
- 3226 views
- $4.00
Related Questions
- Calculus - stationary points, Taylor's series, double integrals..
- Need Upper Bound of an Integral
- Find $\lim\limits _{n\rightarrow \infty} n^2 \prod\limits_{k=1}^{n} (\frac{1}{k^2}+\frac{1}{n^2})^{\frac{1}{n}}$
- Calc limit problem
- Calculus / imaginary numbers and S^2
- Basic calc help
- Find a general solution for $\int at^ne^{bt}dt$, where $n$ is any integer, and $a$ and $b$ are real constants.
- Integral of the fundamentla solution of the heat equation
