← 2017 Paper 1

UPSC 2017 Maths Optional Paper 1 Q8c-ii — Step-by-Step Solution

8 marks · Section B

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

Question

Using Green’s theorem, evaluate the $\displaystyle\int_C \vec F(\vec r)\cdot d\vec r$ counterclockwise where $\vec F(\vec r)=(x^2+y^2)\hat i+(x^2-y^2)\hat j$ and $d\vec r=dx\,\hat i+dy\,\hat j$ and the curve $C$ is the boundary of the region $R=\{(x,y)\mid 1\le y\le 2-x^2\}$.

Technique

Green’s theorem $\oint P\,dx+Q\,dy=\iint(Q_x-P_y)\,dA$ with $Q_x-P_y=2x-2y$ over the parabola–line region.

Solution

Step 1 — Set up Green’s theorem

With $P=x^2+y^2$ and $Q=x^2-y^2$, Green’s theorem gives

Step 2 — Describe the region

$R=\{1\le y\le 2-x^2\}$: bounded below by the horizontal line $y=1$ and above by the parabola $y=2-x^2$. They meet where $2-x^2=1\Rightarrow x^2=1\Rightarrow x=\pm1$. So $x\in[-1,1]$ and, for each $x$, $y$ runs from $1$ to $2-x^2$.

Step 3 — Evaluate the double integral

Inner integral:

The $2x(2-x^2)-2x$ part is odd in $x$ and integrates to $0$ over $[-1,1]$. The remaining even part is

Answer