Application of Integrals — JEE Mains Mathematics

Formulas (3)

• Top minus bottom between the meeting points · Area between two curves
  $$A=\int_{\alpha}^{\beta}\big(f(x)-g(x)\big)\,dx,\qquad\int_{\alpha}^{\beta}a(x-\alpha)(\beta-x)\,dx=\frac{a(\beta-\alpha)^3}{6}$$
• Split where the top or bottom changes · Split at the switch point c
  $$A=\int_{a}^{c}\big(f_1-g\big)\,dx+\int_{c}^{b}\big(f_2-g\big)\,dx$$
• Sideways parabolas as square roots · Under a square-root curve
  $$\int_0^{c}\sqrt{kx}\,dx=\frac23\,c\sqrt{kc}$$

Watch out for (3)

• A third bound can cut the top off
  → Top minus bottom between the meeting points
• A tangent to a cubic crosses it again
  → Split where the top or bottom changes
• The lower branch
  → Sideways parabolas as square roots

Formulas (3)

• Right minus left, integrated in y · Horizontal strips
  $$A=\int_{c}^{d}\big(x_{\text{right}}-x_{\text{left}}\big)\,dy$$
• Two sideways curves, or a choice of strips · Symmetric pair
  $$\int_{-k}^{k}a\,(k^2-y^2)\,dy=\frac{4ak^3}{3}$$
• A tangent to a sideways parabola · Parabola minus tangent
  $$\int_{0}^{y_0}a\,(y-y_0)^2\,dy=\frac{a\,y_0^{3}}{3}$$

Watch out for (3)

• Limits in y, not x
  → Right minus left, integrated in y
• Changing the strips changes both limits and integrand
  → Two sideways curves, or a choice of strips
• dx/dy, not dy/dx
  → A tangent to a sideways parabola

Formulas (3)

• A circle and a parabola · Under a circular arc
  $$\int\sqrt{r^2-x^2}\,dx=\frac{x}{2}\sqrt{r^2-x^2}+\frac{r^2}{2}\sin^{-1}\frac{x}{r}+C$$
• Segments: a circle cut by a line or a circle · Minor segment
  $$\text{segment}=\frac12r^2(\theta-\sin\theta)$$
• Ellipses, hyperbolas and other closed curves · Area of an ellipse
  $$\frac{x^2}{a^2}+\frac{y^2}{b^2}\le1:\quad A=\pi ab$$

Watch out for (3)

• Inside and outside a parabola
  → A circle and a parabola
• Minor or major
  → Segments: a circle cut by a line or a circle
• Standard form first
  → Ellipses, hyperbolas and other closed curves

Formulas (2)

• Moduli of linear terms · Split at the corner
  $$|x-a|=\begin{cases}x-a,&x\ge a\\a-x,&x<a\end{cases}$$
• The modulus of a quadratic · Split at the zeros
  $$\int_p^q|f(x)|\,dx=\sum_{\text{pieces}}\left|\int f(x)\,dx\right|$$

Watch out for (2)

• Each side of a V against a curve
  → Moduli of linear terms
• Where the line cuts the cap
  → The modulus of a quadratic

Formulas (2)

• The min or max of two curves · Integrate the lower curve on each piece
  $$\int_a^b\min\{f,g\}\,dx=\int_a^c f\,dx+\int_c^b g\,dx\quad(f\le g\text{ on }[a,c])$$
• Piecewise rules, composites and the greatest integer · Add the pieces
  $$\int_a^b f\,dx=\sum_{i}\int_{x_{i-1}}^{x_i}f_i(x)\,dx$$

Watch out for (2)

• The min can dip below the axis
  → The min or max of two curves
• Split at every jump
  → Piecewise rules, composites and the greatest integer

Formulas (3)

• Sine and cosine · Between consecutive crossings
  $$\int_{\pi/4}^{5\pi/4}(\sin x-\cos x)\,dx=2\sqrt2$$
• Exponential and log curves · Log and exponential
  $$\int\ln x\,dx=x\ln x-x+C,\qquad\int a^x\,dx=\frac{a^x}{\ln a}+C$$
• Reciprocal curves and xy = k · Under y = k/x
  $$\int_a^b\frac{k}{x}\,dx=k\ln\frac{b}{a}$$

Watch out for (3)

• Area is not the signed integral
  → Sine and cosine
• The ln a factor
  → Exponential and log curves
• State the quadrant
  → Reciprocal curves and xy = k

Formulas (2)

• Area in terms of the constant, then solve · Parabola and a line through its vertex
  $$y=ax^2,\ y=mx:\quad A=\frac{m^3}{6a^2}$$
• Dividing an area, and making it largest or smallest · A piece in a given ratio
  $$A_1=\frac{m}{m+n}\,A_{\text{total}}$$

Watch out for (2)

• Keep the sign condition
  → Area in terms of the constant, then solve
• Which piece comes first
  → Dividing an area, and making it largest or smallest