PYQ Vault

JEE Mains Maths · Formula sheet

Application of Integrals formulas

18 formulas and 18 common traps for JEE Mains Maths Application of Integrals, grouped by subtopic.

Full notes with worked examples

Curves and Lines: Vertical Strips

Learn this subtopic in the notes

Top minus bottom between the meeting points

Area between two curves

A=∫αβ(f(x)−g(x)) dx,∫αβa(x−α)(β−x) dx=a(β−α)36A=\int_{\alpha}^{\beta}\big(f(x)-g(x)\big)\,dx,\qquad\int_{\alpha}^{\beta}a(x-\alpha)(\beta-x)\,dx=\frac{a(\beta-\alpha)^3}{6}

Split where the top or bottom changes

Split at the switch point c

A=∫ac(f1−g) dx+∫cb(f2−g) dxA=\int_{a}^{c}\big(f_1-g\big)\,dx+\int_{c}^{b}\big(f_2-g\big)\,dx

Sideways parabolas as square roots

Under a square-root curve

∫0ckx dx=23 ckc\int_0^{c}\sqrt{kx}\,dx=\frac23\,c\sqrt{kc}

Common traps

A third bound can cut the top off

When the region also has y≤cy\le c, the top is the lower of the two upper bounds. Find where the cap meets each curve and split the interval there before integrating.

A tangent to a cubic crosses it again

Solve curve = tangent in full. The point of contact is a double root, and the other root is where the tangent crosses the cubic again — that is the second limit.

The lower branch

y2=kxy^2=kx also has the branch y=−kxy=-\sqrt{kx}. If the region is not limited to y≥0y\ge0, include the part below the x-axis, or use symmetry and double.

Horizontal Strips and Sideways Parabolas

Learn this subtopic in the notes

Right minus left, integrated in y

Horizontal strips

A=∫cd(xright−xleft) dyA=\int_{c}^{d}\big(x_{\text{right}}-x_{\text{left}}\big)\,dy

Two sideways curves, or a choice of strips

Symmetric pair

∫−kka (k2−y2) dy=4ak33\int_{-k}^{k}a\,(k^2-y^2)\,dy=\frac{4ak^3}{3}

A tangent to a sideways parabola

Parabola minus tangent

∫0y0a (y−y0)2 dy=a y033\int_{0}^{y_0}a\,(y-y_0)^2\,dy=\frac{a\,y_0^{3}}{3}

Common traps

Limits in y, not x

Solve for the y-values where the curves meet. Substituting to get x-values first, and then using those as limits of a y-integral, gives a wrong area.

Changing the strips changes both limits and integrand

When a question rewrites an x-integral as a y-integral, redraw the region and read each piece's left and right boundary afresh. Swapping only the limits, or only the integrand, gives a different region.

dx/dy, not dy/dx

For a sideways parabola, dxdy\frac{dx}{dy} gives the tangent directly as x in terms of y. If you use dydx\frac{dy}{dx}, its slope is the reciprocal — mixing the two gives a line that does not touch the curve.

Circles, Ellipses and Other Conics

Learn this subtopic in the notes

A circle and a parabola

Under a circular arc

∫r2−x2 dx=x2r2−x2+r22sin⁡−1xr+C\int\sqrt{r^2-x^2}\,dx=\frac{x}{2}\sqrt{r^2-x^2}+\frac{r^2}{2}\sin^{-1}\frac{x}{r}+C

Segments: a circle cut by a line or a circle

Minor segment

segment=12r2(θ−sin⁡θ)\text{segment}=\frac12r^2(\theta-\sin\theta)

Ellipses, hyperbolas and other closed curves

Area of an ellipse

x2a2+y2b2≤1:A=πab\frac{x^2}{a^2}+\frac{y^2}{b^2}\le1:\quad A=\pi ab

Common traps

Inside and outside a parabola

Inside y2=kxy^2=kx means y2≤kxy^2\le kx, the side that holds the focus. Outside is the rest of the circle. Sketch both before deciding which piece to subtract from which.

Minor or major

The formula gives the smaller segment. When the question asks for the larger portion, subtract it from πr2\pi r^2 — and a V-shaped cut needs the two pieces on each side added.

Standard form first

4x2+9y2=364x^2+9y^2=36 is x29+y24=1\frac{x^2}9+\frac{y^2}4=1, so a=3a=3, b=2b=2 and the area is 6π6\pi. Reading a and b off the unscaled equation gives a wrong area.

Modulus Curves

Learn this subtopic in the notes

Moduli of linear terms

Split at the corner

∣x−a∣={x−a,x≥aa−x,x<a|x-a|=\begin{cases}x-a,&x\ge a\\a-x,&x<a\end{cases}

The modulus of a quadratic

Split at the zeros

∫pq∣f(x)∣ dx=∑pieces∣∫f(x) dx∣\int_p^q|f(x)|\,dx=\sum_{\text{pieces}}\left|\int f(x)\,dx\right|

Common traps

Each side of a V against a curve

When a V meets a parabola or x\sqrt x, its two arms meet the curve at different points. Solve each arm separately; one equation for both sides loses a limit.

Where the line cuts the cap

If the line y=cy=c sits below the top of the folded cap, the cap pokes through and makes a second region above the line. Check whether c is above or below the cap's peak before deciding how many regions there are.

Max, Min and Piecewise Boundaries

Learn this subtopic in the notes

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

Common traps

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.

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.

Trigonometric, Exponential and Reciprocal Curves

Learn this subtopic in the notes

Sine and cosine

Between consecutive crossings

∫π/45π/4(sin⁡x−cos⁡x) dx=22\int_{\pi/4}^{5\pi/4}(\sin x-\cos x)\,dx=2\sqrt2

Exponential and log curves

Log and exponential

∫ln⁡x dx=xln⁡x−x+C,∫ax dx=axln⁡a+C\int\ln x\,dx=x\ln x-x+C,\qquad\int a^x\,dx=\frac{a^x}{\ln a}+C

Reciprocal curves and xy = k

Under y = k/x

∫abkx dx=kln⁡ba\int_a^b\frac{k}{x}\,dx=k\ln\frac{b}{a}

Common traps

Area is not the signed integral

∫02πsin⁡x dx=0\int_0^{2\pi}\sin x\,dx=0, but the area is 4. Split wherever the curve crosses the axis, or where the top and bottom swap, and add the pieces as positive numbers.

The ln a factor

∫012x dx=1ln⁡2\int_0^1 2^x\,dx=\frac1{\ln2}, not 1. Only base e integrates to itself; any other base divides by its log.

State the quadrant

Without x≥0x\ge0, a region such as xy≤kxy\le k, 1≤y≤x21\le y\le x^2 includes every x≤−1x\le-1, where xy≤0≤kxy\le0\le k always holds, so it is unbounded. The printed answers assume x≥0x\ge0.

Unknown Parameters and Area Ratios

Learn this subtopic in the notes

Area in terms of the constant, then solve

Parabola and a line through its vertex

y=ax2, y=mx:A=m36a2y=ax^2,\ y=mx:\quad A=\frac{m^3}{6a^2}

Dividing an area, and making it largest or smallest

A piece in a given ratio

A1=mm+n AtotalA_1=\frac{m}{m+n}\,A_{\text{total}}

Common traps

Keep the sign condition

A square root gives two values of the constant; the condition in the question (a>0a>0, α>0\alpha>0) keeps one. Check the kept value also makes the curves meet as the region needs.

Which piece comes first

A ratio m:nm:n depends on which piece the question names first. Compute both pieces and match the order in the question before reading off m and n.

More JEE Mains Maths formula sheets