Question

If C is the rectangle bounded by the lines $x = 0, x = a, y = 0, y = b$, then the value of $\int_{C}(x^{2} + y^{2})dx - 2xy dy$ is}

• A. $2ab^{2}$
• B. $-2ab^{2}$
• C. $ab^{2}$
• D. $-ab^{2}$

Hint:

Green's Theorem is your best friend for line integrals around a closed rectangle or circle.

Answer 1

The Correct Option is B

Step 1: Concept
By Green's Theorem, $\oint_{C} M dx + N dy = \iint_{R} (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}) dA$.

Step 2: Meaning
$M = x^{2} + y^{2}$ and $N = -2xy$. Calculate partial derivatives: $\frac{\partial N}{\partial x} = -2y$ and $\frac{\partial M}{\partial y} = 2y$.

Step 3: Analysis
The integrand is $\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = -2y - 2y = -4y$. We integrate this over the rectangle: $\int_{0}^{a} \int_{0}^{b} -4y dy dx$.

Step 4: Conclusion
$\int_{0}^{a} [-2y^{2}]_{0}^{b} dx = \int_{0}^{a} -2b^{2} dx = -2ab^{2}$. Final Answer: (B)