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UPSC 2016 Maths Optional Paper 1 Q4c — Step-by-Step Solution

15 marks · Section A

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

Question

Evaluate $\displaystyle\iint_R f(x,y)\,dx\,dy$ over the rectangle $R=[0,1;0,1]$ where

Technique

Integrate the indicator-restricted function by describing the region as $x^2<y<2x^2$, capping the upper curve at $y=1$ where it exits the square; split the $x$-range at $2x^2=1$.

Solution

Step 1 — Describe the integration region inside the unit square

The integrand is nonzero only where $x^2<y<2x^2$. Inside $[0,1]^2$, for a fixed $x\in[0,1]$ the band runs from $y=x^2$ up to $y=2x^2$, but $y$ cannot exceed $1$.

• If $2x^2\le1$, i.e. $x\le\tfrac{1}{\sqrt2}$: the full band $x^2<y<2x^2$ lies in the square.
• If $x>\tfrac{1}{\sqrt2}$: the upper limit is capped at $y=1$ (since $2x^2>1$), and the band is $x^2<y<1$ (valid while $x^2<1$, i.e. all $x<1$).

Step 2 — Split the integral at $x=\tfrac{1}{\sqrt2}$

Step 3 — Inner integrals

$\displaystyle\int (x+y)\,dy=xy+\tfrac{y^2}{2}$.

For $I_1$ (from $x^2$ to $2x^2$):

For $I_2$ (from $x^2$ to $1$):

Step 4 — Total

Answer