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JEE Mains Maths · Application of Integrals

Max, Min and Piecewise Boundaries

Regions whose boundary is the smaller or larger of two curves, or a function defined in pieces, split wherever the rule changes.

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Why this matters

Eleven PYQs, four of them multiple choice. Seven have a boundary given by the min or max of two curves, so the top switches where they cross; four define the boundary piecewise — through a composite, a greatest-integer term, a sign chart or a differentiability condition. Two ideas cover the page.

Concept 1 of 2: The min or max of two curves

min⁡{f,g}\min\{f,g\} is whichever curve is lower at each x, so it switches from one to the other where f=gf=g. Find those crossings, decide which curve is lower on each piece, and integrate that one. 0≤y≤min⁡{f,g}0\le y\le\min\{f,g\} is simply the region under both curves at once.

Definition

  • min⁡{f,g}=f\min\{f,g\}=f where f≤gf\le g, and g elsewhere; max⁡\max is the reverse.
  • The switch points are the roots of f=gf=g.
  • With 0≤y0\le y, keep only the x where the boundary is non-negative.

Integrate the lower curve on each piece

∫abmin⁡{f,g} dx=∫acf dx+∫cbg dx(f≤g on [a,c])\int_a^b\min\{f,g\}\,dx=\int_a^c f\,dx+\int_c^b g\,dx\quad(f\le g\text{ on }[a,c])

Worked example

Find the area of {(x,y):0≤x≤3, 0≤y≤min⁡{x2,4}}\{(x,y):0\le x\le3,\ 0\le y\le\min\{x^2,4\}\}.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 28 Jan 2025 · Q70Moderate

Example 1 · Application of Integrals · Max, Min and Piecewise Boundaries

The area (in sq. units) of the region {(x,y):0≤y≤2∣x∣+1,0≤y≤x2+1,∣x∣≤3}\left\{ (x,y):0 \leq y \leq 2|x| + 1,0 \leq y \leq x^{2}+ 1,|x| \leq 3 \right\} is

The min can dip below the axis

With 0≤y≤min⁡{f,g}0\le y\le\min\{f,g\}, the region exists only where the min is non-negative. Find where the lower curve crosses the x-axis — that, not a crossing of f and g, may be the end of the region.
Drill 6 more on the min or max of two curves

Concept 2 of 2: Piecewise rules, composites and the greatest integer

Rewrite the boundary as a list of ordinary curves on intervals, then integrate piece by piece. For a composite, find the pieces of the inner function first and apply the outer one to each. For [x][x], each unit interval is a constant. For a sign condition, draw the sign chart and keep only the intervals that pass.

Definition

  • [x]=n[x]=n on [n,n+1)[n,n+1).
  • t+∣t∣2\frac{t+|t|}2 is t for t≥0t\ge0 and 0 for t<0t<0.
  • Differentiable at a joint: the two pieces agree in value and in slope there.
  • Sign condition: keep the intervals of the sign chart where it holds.

Add the pieces

∫abf dx=∑i∫xi−1xifi(x) dx\int_a^b f\,dx=\sum_{i}\int_{x_{i-1}}^{x_i}f_i(x)\,dx

Worked example

Let f(t)=t+∣t∣2f(t)=\frac{t+|t|}2. Find the area between y=f(x2−1)y=f(x^2-1) and the x-axis for −2≤x≤2-2\le x\le2.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 22 Jan 2025 · Q71Moderate

Example 2 · Application of Integrals · Max, Min and Piecewise Boundaries

Let the function,
f(x)={−3ax2−2,x<1a2+bx,x≥1f(x) =\left\{ \begin{matrix} - 3ax^{2}- 2, & x < 1 \\ a^{2}+ bx, & x \geq 1 \end{matrix} \right.
Be differentiable for all x∈Rx \in R, where a>1, b∈Ra > 1,\text{ }b \in R. If the area of the region enclosed by y=f(x)y = f(x) and the line y=−20y = - 20 is α+β3,α,β,∈Z\alpha + \beta\sqrt{3},\alpha,\beta, \in Z, then the value of α+β\alpha + \beta is ________\_\_\_\_\_\_\_\_ .

Split at every jump

[x2][x^2] jumps where x2x^2 is an integer: at x=1x=1, 2\sqrt2 and 3\sqrt3, not only at whole numbers of x. Find where the inside of the bracket crosses an integer.
Drill 3 more on piecewise rules, composites and the greatest integer

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • The min or max of two curves

    Integrate the lower curve on each piece

    ∫abmin⁡{f,g} dx=∫acf dx+∫cbg dx(f≤g on [a,c])\int_a^b\min\{f,g\}\,dx=\int_a^c f\,dx+\int_c^b g\,dx\quad(f\le g\text{ on }[a,c])
  • Piecewise rules, composites and the greatest integer

    Add the pieces

    ∫abf dx=∑i∫xi−1xifi(x) dx\int_a^b f\,dx=\sum_{i}\int_{x_{i-1}}^{x_i}f_i(x)\,dx

Watch out for (2)

Test yourself on Application of Integrals

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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