The definite integral as the limit of a sum

What you will be able to do

• write the area under a curve as a limit of sums of rectangles;
• use $\sum r$, $\sum r^2$ and $\sum r^3$ to evaluate an integral from the definition;
• turn a limit of a sum of the form $\dfrac{1}{n}\sum f\left(\dfrac{r}{n}\right)$ into an integral.

Counting rectangles

Before the Fundamental Theorem, mathematicians found areas by adding rectangles. Some Indian textbooks now treat this method briefly. A-Level, IB and AP courses use Riemann sums, which are these sums of rectangles, and so do JEE Main questions. Cut $[a, b]$ into $n$ strips of equal width $h = \dfrac{b - a}{n}$. On each strip draw a rectangle whose height is the value of $f$ at the right end of the strip. The right ends are $a + h$, $a + 2h$, $\dots$, $a + nh = b$.

The total area of the rectangles is the right-hand sum. It is $h$ times the total of the heights $f(a + h)$, $f(a + 2h)$, and so on up to $f(a + nh)$. In symbols,

When $n$ grows, $h$ shrinks, and the rectangles hug the curve more closely. For a continuous function the sums tend to a limit, and that limit is the definite integral:

Using the left ends instead, or the middle of each strip, gives the same limit.

Sums you need

Also $\sum_{r=1}^{n} 1 = n$, and for a geometric sum, $\sum_{r=0}^{n-1} q^r = \dfrac{q^n - 1}{q - 1}$.

Worked examples

Example 1. Find $\displaystyle\int_0^2 (x^2 + 1)\,dx$ from the definition.

1. Here $a = 0$, $b = 2$ and $h = \dfrac{2}{n}$. The $r$-th height is $f(rh) = r^2h^2 + 1$.
2. The sum is $S_n = h\displaystyle\sum_{r=1}^{n}(r^2h^2 + 1) = h^3 \times \dfrac{n(n + 1)(2n + 1)}{6} + hn$.
3. With $h = \dfrac{2}{n}$ this is $\dfrac{8}{n^3} \times \dfrac{n(n + 1)(2n + 1)}{6} + 2 = \dfrac{4}{3}\left(1 + \dfrac{1}{n}\right)\left(2 + \dfrac{1}{n}\right) + 2$.
4. As $n \to \infty$, $\dfrac{1}{n} \to 0$. The limit is $\dfrac{4}{3} \times 2 + 2 = \dfrac{14}{3}$.
5. Check with the theorem: $\left[\dfrac{x^3}{3} + x\right]_0^2 = \dfrac{8}{3} + 2 = \dfrac{14}{3}$.

Example 2. Find $\displaystyle\int_1^3 (2x + 1)\,dx$ from the definition.

1. Here $a = 1$ and $h = \dfrac{2}{n}$. The $r$-th height is $f(1 + rh) = 2(1 + rh) + 1 = 3 + 2rh$.
2. $S_n = h\displaystyle\sum_{r=1}^{n}(3 + 2rh) = 3hn + 2h^2 \times \dfrac{n(n + 1)}{2} = 6 + \dfrac{4(n + 1)}{n}$.
3. As $n \to \infty$, $S_n \to 6 + 4 = 10$. This agrees with the theorem.

Example 3 (a limit of a sum, read as an integral). Find $\displaystyle\lim_{n \to \infty}\dfrac{1}{n}\sum_{r=1}^{n}\left(\dfrac{r}{n}\right)^3$.

1. This is a right-hand sum on $[0, 1]$ with $h = \dfrac{1}{n}$ and $f(x) = x^3$, because $a + rh = \dfrac{r}{n}$.
2. The limit is $\displaystyle\int_0^1 x^3\,dx = \dfrac{1}{4}$.

To turn a limit of a sum into an integral, write the term as $\dfrac{1}{n} f\left(\dfrac{r}{n}\right)$. Then $\dfrac{r}{n}$ becomes $x$, $\dfrac{1}{n}$ becomes $dx$, and the sum from $r = 1$ to $n$ becomes the integral from $0$ to $1$.

Practise with new numbers

Try it in Python

The program adds right-hand rectangles under $y = x^2$ from $0$ to $3$, for larger and larger $n$. The exact area is $9$.

Common mistakes

• Using the wrong width. $h = \dfrac{b - a}{n}$. For $[1, 3]$ it is $\dfrac{2}{n}$, not $\dfrac{3}{n}$.
• Forgetting the shift $a$. The heights are $f(a + rh)$, and for $a \ne 0$ the term $a$ must stay.
• Wrong sum formulas. $\sum r^2 = \dfrac{n(n + 1)(2n + 1)}{6}$, which has $2n + 1$, not $2n$.
• Not taking the limit. After finding $S_n$, let $\dfrac{1}{n} \to 0$.
• Wrong limits when reading a sum as an integral. If $r$ runs from $1$ to $n$ and the term is $\dfrac{1}{n}f\left(\dfrac{r}{n}\right)$, the integral runs from $0$ to $1$.

Remember these

• $\int_a^b f(x)\,dx = \lim_{n \to \infty} h\sum_{r=1}^{n} f(a + rh)$ with $h = \dfrac{b - a}{n}$.
• $\sum r = \dfrac{n(n + 1)}{2}$, $\sum r^2 = \dfrac{n(n + 1)(2n + 1)}{6}$, $\sum r^3 = \left[\dfrac{n(n + 1)}{2}\right]^2$.
• $\lim \dfrac{1}{n}\sum_{r=1}^{n} f\left(\dfrac{r}{n}\right) = \int_0^1 f(x)\,dx$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Use the definition to find $\displaystyle\int_0^1 x\,dx$.

2. What is the width $h$ of each strip when $[2, 5]$ is cut into $n$ strips? What are the first and last right-hand ends?

3. Write $\displaystyle\lim_{n \to \infty}\dfrac{1}{n}\sum_{r=1}^{n}\left(\dfrac{r}{n}\right)^2$ as an integral and find it.

4. The right-hand sum for $\displaystyle\int_0^1 e^x\,dx$ with $n$ strips is $\dfrac{1}{n}\sum_{r=1}^{n}e^{r/n}$. Is it more or less than the exact value $e - 1$?

5. For $f(x) = x$ on $[0, 2]$ with $n = 4$, find the right-hand sum.

References

• OpenStax, Calculus Volume 1, Chapter 5, sections 5.1 Approximating Areas and 5.2 The Definite Integral.
• NCERT, Mathematics Part II for Class XII, Chapter 7, the section on the definite integral.