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UPSC 2015 Maths Optional Paper 1 Q3d — Step-by-Step Solution

12 marks · Section A

Double integrals · Calculus · asked 10× in 13 yrs · Read the full method →

Question

Evaluate $\displaystyle\iint_R (x-y)^2\cos^2(x+y)\,dx\,dy$, where $R$ is the rhombus with successive vertices $(\pi,0)$, $(2\pi,\pi)$, $(\pi,2\pi)$, $(0,\pi)$.

Technique

Affine change of variables $u=x+y,\,v=x-y$ transforms a rhombus aligned with $x\pm y=$ const into an axis-aligned rectangle; integrand factorises.

Solution

Strategy. The integrand depends on $x-y$ and $x+y$. The rhombus is bounded by lines $x+y=\text{const}$ and $x-y=\text{const}$ (sides connect consecutive vertices). So substitute $u=x+y$, $v=x-y$.

Step 1 — Identify the sides of the rhombus

Successive vertices: $A=(\pi,0)$, $B=(2\pi,\pi)$, $C=(\pi,2\pi)$, $D=(0,\pi)$.

• $AB$: from $(\pi,0)$ to $(2\pi,\pi)$. $x-y$ along: $\pi$ (constant). So $x-y=\pi$.
• $BC$: from $(2\pi,\pi)$ to $(\pi,2\pi)$. $x+y=3\pi$ (constant).
• $CD$: from $(\pi,2\pi)$ to $(0,\pi)$. $x-y=-\pi$ (constant).
• $DA$: from $(0,\pi)$ to $(\pi,0)$. $x+y=\pi$ (constant).

So in $(u,v)=(x+y,x-y)$ coordinates, $R$ is the rectangle $[\pi,3\pi]\times[-\pi,\pi]$.

Step 2 — Change of variables

$u=x+y$, $v=x-y$. Then $x=(u+v)/2$, $y=(u-v)/2$.

Jacobian $\dfrac{\partial(x,y)}{\partial(u,v)}=\det\begin{pmatrix}1/2 & 1/2\\ 1/2 & -1/2\end{pmatrix}=-1/2$, so $|J|=1/2$.

$dx\,dy=\tfrac{1}{2}\,du\,dv$.

Step 3 — Rewrite integral

Step 4 — Evaluate each integral

Step 5 — Combine

Answer