The 1/6 formula

If a parabola $y=ax^2+bx+c$ and a line meet at $x=\alpha$ and $x=\beta$ ($\alpha<\beta$), their difference factors as $a(x-\alpha)(x-\beta)$. The integral formula above then gives the enclosed area.

Here $a$ is the $x^2$ coefficient of the difference. For $y=4x^2-4x$ and $y=4x$, the difference is $4x^2-8x=4x(x-2)$, so $a=4$, $\alpha=0$, $\beta=2$, and $S=\frac{4}{6}\cdot 2^3=\frac{16}{3}$.

Examples

1. 01You found $a=2$ and $\beta-\alpha=4$. What is the area?$\displaystyle \text{area} = \frac{|a|}{6}(\beta-\alpha)^3$AnswerHide

   Answer$\displaystyle \frac{64}{3}$

   Steps

   1. $\displaystyle \frac{2}{6}\cdot4^3$$\displaystyle = \frac{128}{6}$$\displaystyle = \frac{64}{3}$
2. 02In area $=\frac{|a|}{6}(\beta-\alpha)^3$, what is $a$?$\displaystyle y=5x^2-2x,\quad y=2x$AnswerHide

   Answer$\displaystyle 5$

   Steps

   1. $\displaystyle 5x^2-2x = 2x$$\displaystyle \Rightarrow 5x^2-4x = 0$
   2. Read the coefficient of x²
   3. $\displaystyle a$$\displaystyle = 5$
3. 03In area $=\frac{|a|}{6}(\beta-\alpha)^3$, what is $a$?$\displaystyle y=2x^2-x,\quad y=x$AnswerHide

   Answer$\displaystyle 2$

   Steps

   1. $\displaystyle 2x^2-x = x$$\displaystyle \Rightarrow 2x^2-2x = 0$
   2. Read the coefficient of x²
   3. $\displaystyle a$$\displaystyle = 2$
4. 04In area $=\frac{|a|}{6}(\beta-\alpha)^3$, what is $a$?$\displaystyle y=4x^2-5x,\quad y=5x$AnswerHide

   Answer$\displaystyle 4$

   Steps

   1. $\displaystyle 4x^2-5x = 5x$$\displaystyle \Rightarrow 4x^2-10x = 0$
   2. Read the coefficient of x²
   3. $\displaystyle a$$\displaystyle = 4$
5. 05In area $=\frac{|a|}{6}(\beta-\alpha)^3$, what is $a$?$\displaystyle y=2x^2-3x,\quad y=3x$AnswerHide

   Answer$\displaystyle 2$

   Steps

   1. $\displaystyle 2x^2-3x = 3x$$\displaystyle \Rightarrow 2x^2-6x = 0$
   2. Read the coefficient of x²
   3. $\displaystyle a$$\displaystyle = 2$
6. 06In area $=\frac{|a|}{6}(\beta-\alpha)^3$, what is $a$?$\displaystyle y=3x^2-4x,\quad y=4x$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. $\displaystyle 3x^2-4x = 4x$$\displaystyle \Rightarrow 3x^2-8x = 0$
   2. Read the coefficient of x²
   3. $\displaystyle a$$\displaystyle = 3$

FAQ

Q1Is $a$ from the line or from the parabola?AnswerHide

A

It is the $x^2$ coefficient of the difference. For a parabola and a line, that equals the parabola's $x^2$ coefficient. For two parabolas, $a$ is the difference of their $x^2$ coefficients.

Q2When does it not work?AnswerHide

A

When the difference is not quadratic. For example, the region between a cubic curve and a line gives a cubic difference, so the 1/6 formula does not apply directly.