← 2025 Paper 1

UPSC 2025 Maths Optional Paper 1 Q7b — Step-by-Step Solution

15 marks · Section B

Line integrals · Vector Analysis · asked 8× in 14 yrs · Read the full method →

Question

Verify Green’s theorem in the plane for $\displaystyle\oint_C [(xy + y^2)\,dx + x^2\,dy]$, where $C$ is the boundary of the region bounded by the curves $y = x$ and $y = x^2$.

Technique

Apply Green’s theorem $\displaystyle\oint_C (P\,dx + Q\,dy) = \iint_R\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA$ by computing both sides independently and checking they agree.

Solution

Here $P = xy + y^2$, $Q = x^2$. The curves $y=x$ and $y=x^2$ intersect where $x = x^2$, i.e. at $(0,0)$ and $(1,1)$. On $[0,1]$, $x \ge x^2$, so the region is

The positively-oriented (counterclockwise) boundary $C$ consists of:

• $C_1$: the parabola $y=x^2$ from $(0,0)$ to $(1,1)$ (lower boundary, left→right);
• $C_2$: the line $y=x$ from $(1,1)$ back to $(0,0)$ (upper boundary, right→left).

Double integral (RHS)

Inner integral:

Then

Line integral (LHS)

Along $C_1$: $y = x^2$, $dy = 2x\,dx$, $x:0\to1$.

Along $C_2$: $y = x$, $dy = dx$, $x:1\to0$.

Sum:

Conclusion

Green’s theorem is verified.

Answer