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UPSC 2018 Maths Optional Paper 1 Q8c — Step-by-Step Solution

13 marks · Section B

Line integrals · Vector Analysis · asked 8× in 13 yrs · Read the full method →

Question

Let $\vec F=xy^2\hat i+(y+x)\hat j$. Integrate $(\nabla\times\vec F)\cdot\hat k$ over the region in the first quadrant bounded by the curves $y=x^2$ and $y=x$ using Green’s theorem.

Technique

$(\nabla\times\vec F)\cdot\hat k=Q_x-P_y=1-2xy$; integrate over $R$ with $x^2\le y\le x$, $0\le x\le1$.

Solution

Step 1 — The integrand $(\nabla\times\vec F)\cdot\hat k$

With $\vec F=(P,Q)=(xy^2,\ y+x)$ (a planar field),

By Green’s theorem this double integral equals the boundary line integral $\oint_{\partial R}(P\,dx+Q\,dy)$; we evaluate the double integral directly (both are computed below).

Step 2 — Describe the region

The curves $y=x^2$ and $y=x$ meet where $x^2=x$, i.e. $x=0$ and $x=1$. On $[0,1]$ the line lies above the parabola ($x\ge x^2$), so

Step 3 — Iterated integral

Inner integral:

Outer integral:

Common denominator $12$: $\dfrac{6-4-3+2}{12}=\dfrac{1}{12}.$

Answer