Approximating area with rectangles

Splitting $a\le x\le b$ into $n$ equal parts gives rectangles of width $\Delta x=\dfrac{b-a}{n}$. Taking each height as $f(x)$ at the left endpoint, the sum of the rectangle areas approximates the area under the curve.

For example, splitting $y=x^2$, $0\le x\le 1$ into 4 parts gives width $\dfrac14$ and left endpoints $x=0,\ \dfrac14,\ \dfrac12,\ \dfrac34$. The 4th height is $\left(\dfrac34\right)^2=\dfrac{9}{16}$, and the sum is

As the rectangles get thinner, the sum approaches the area under the curve (here $\dfrac13$). This is the idea behind the definite integral.

Examples

1. 01Split into $4$ equal parts and take heights at the left endpoints. What is the height of the $4$th rectangle?$\displaystyle y=x^{2}, \quad 0 \le x \le 1$AnswerHide

   Answer$\displaystyle \frac{9}{16}$

   Steps

   1. $\displaystyle x$$\displaystyle = \frac{3}{4}$
   2. The height there is
   3. $\displaystyle y$$\displaystyle = \left(\frac{3}{4}\right)^2 = \frac{9}{16}$
2. 02Split this interval into $4$ equal parts. What is the width of one rectangle?$\displaystyle 0 \le x \le 3$AnswerHide

   Answer$\displaystyle \frac{3}{4}$

   Steps

   1. $\displaystyle \frac{3 - 0}{4}$$\displaystyle = \frac{3}{4}$
3. 03Split this interval into $3$ equal parts. What is the width of one rectangle?$\displaystyle 0 \le x \le 6$AnswerHide

   Answer$\displaystyle 2$

   Steps

   1. $\displaystyle \frac{6 - 0}{3}$$\displaystyle = 2$
4. 04Split this interval into $3$ equal parts. What is the width of one rectangle?$\displaystyle 0 \le x \le 4$AnswerHide

   Answer$\displaystyle \frac{4}{3}$

   Steps

   1. $\displaystyle \frac{4 - 0}{3}$$\displaystyle = \frac{4}{3}$
5. 05Split this interval into $6$ equal parts. What is the width of one rectangle?$\displaystyle 0 \le x \le 2$AnswerHide

   Answer$\displaystyle \frac{1}{3}$

   Steps

   1. $\displaystyle \frac{2 - 0}{6}$$\displaystyle = \frac{1}{3}$

FAQ

Q1Should the height come from the left or right endpoint?AnswerHide

A

Either one approaches the same area as the rectangles get thinner (for a continuous function). With a coarse split the values differ: for an increasing function like $y=x^2$, left endpoints give a smaller value and right endpoints a larger one.

Q2With 4 parts and left endpoints, where is the left edge of the 4th rectangle?AnswerHide

A

At $x=a+3\Delta x$. The 1st rectangle starts at $x=a$ ($k=0$), so the 4th is $k=3$. For $0\le x\le 1$ that is $x=\dfrac34$.