Riemann Integral

The Riemann Integral is one of the most essential concepts in calculus and real analysis. It provides a rigorous way to define the area under a curve and serves as the foundation for more advanced topics in analysis and applied mathematics. Whether you’re preparing for competitive exams like JEE or diving into Real Analysis, mastering the Riemann integral is a must.

1.0Riemann Integral Definition

The Riemann Integral of a function f(x) over an interval [a, b] is the limit of the sum of areas of rectangles under the curve as the width of the rectangles approaches zero.

Formally, if $f:[a,b]\to ⊠$ is a bounded function, and we divide [a, b] into n subintervals:

$a=x_0<x_1<x_2<\dots <x_n=b$ 

Then for sample points $x_i^{\ast }\in \left[x_{i-1},x_i\right]$, the Riemann sum is:

$S={\sum }_{i=1}^{n}f\left(x_i^{\ast }\right)\Delta x_i$

where $\Delta x_i=x_i-x_{i-1}$. If the limit of these sums exists as $\max \left(\Delta x_i\right)\to 0$, it is called the Riemann Integral:

${\int }_a^{b}f(x)dx$ 

2.0Riemann Integral Formula

The general Riemann Integral formula is:

${\int }_a^{b}f(x)dx={\lim }_{n\to \infty }{\sum }_{i=1}^{n}f\left(x_i^{\ast }\right)\Delta x_i$

Proof of Riemann Integral for f(x) = x over [0, 1]

Let’s divide [0, 1] into n equal parts, each of width $\Delta x=\frac{1}{n}$, and use right endpoints:

$x_i=\frac{i}{n}$

$S_n={\sum }_{i=1}^n\frac{i}{n}\cdot \frac{1}{n}=\frac{1}{n^2}{\sum }_{i=1}^{n}i=\frac{1}{n^2}\cdot \frac{n(n+1)}{2}=\frac{n+1}{2n}$ 

As $n\to \infty ,S_n\to \frac{1}{2}$. Hence:

${\int }_0^{1}xdx=\frac{1}{2}$

3.0Riemann Integral in Real Analysis 

In Real Analysis, the Riemann Integral is used to study the integration of bounded functions and understand properties like continuity, limits, and measure. It leads to more advanced theories like the Lebesgue Integral for handling more complex functions.

4.0Riemann Integral Properties

1. Linearity:

${\int }_a^b(af(x)+bg(x))dx=a{\int }_a^{b}f(x)dx+b{\int }_a^{b}g(x)dx$

2. Additivity over intervals:

${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$ 

3. Monotonicity:

$\begin{array}{rl}& \text{ If }f(x)\le g(x)\text{ on }[\mathrm{a},\mathrm{b}]\text{, then }\\ & {\int }_a^{b}f(x)dx\le {\int }_a^{b}g(x)dx\end{array}$

4. Non-negativity:

$\text{ If }f(x)\ge 0\text{ on [a, b], then }{\int }_a^{b}f(x)dx\ge 0$

5.0Upper and Lower Riemann Integral

• Lower Sum (L(f,P)): Uses the minimum value of the function on each subinterval.
• Upper Sum (U(f,P)): Uses the maximum value of the function on each subinterval.

If the supremum of all lower sums equals the infimum of all upper sums, the function is Riemann integrable.

6.0Solved Examples on Riemann Integral

Example 1: Evaluate: ${\int }_1^4\sqrt{x}dx$ 

Solution:

${\int }_1^4\sqrt{x}dx={\left[\frac{2}{3}{x}^{3/2}\right]}_1^4=\frac{2}{3}(8-1)=\frac{14}{3}$ 

Example 2: If

$f(x)=x^2+2x,\text{ find }{\int }_{-1}^{2}f(x)dx$

Solution:

$\begin{array}{rl}& {\int }_{-1}^2\left(x^2+2x\right)dx\\ & ={\left[\frac{x^3}{3}+x^2\right]}_{-1}^2\\ & =\left(\frac{8}{3}+4\right)-\left(-\frac{1}{3}+1\right)\\ & =\frac{19}{3}-\frac{2}{3}=\frac{17}{3}\end{array}$

Example 3: Evaluate the Riemann Integral of $f(x)=x^2$ over the interval [1, 3].

Solution:

The integral we want to evaluate is:

${\int }_1^3{x}^{2}dx$ 

Step 1: Find the antiderivative of $f(x)=x^2$.

The antiderivative of x² is $\frac{x^3}{3}$.

Step 2: Apply the principles of the Fundamental Theorem of Calculus.

Now, we evaluate the integral by applying the limits:

$\begin{array}{rl}& {\int }_1^3{x}^{2}dx={\left[\frac{x^3}{3}\right]}_1^3\\ & =\frac{3^3}{3}-\frac{1^3}{3}\\ & =\frac{27}{3}-\frac{1}{3}=\frac{26}{3}\end{array}$ 

Thus, the value of the Riemann integral is $\frac{26}{3}$ 

Example 4: Evaluate the Riemann Integral of the piecewise function 

$f(x)=\{\begin{array}{ll}x& \text{ if }0\le x\le 2\\ 4-x& \text{ if }2<x\le 4\end{array}$ 

over the interval [0, 4].

Solution:

We need to evaluate the integral of f(x) over the interval [0, 4]. Since f(x) is piecewise, we split the integral into two parts:

${\int }_0^{4}f(x)dx={\int }_0^{2}xdx+{\int }_2^4(4-x)dx$

Step 1: Evaluate the first integral ${\int }_0^{2}xdx$.

The antiderivative of x is $\frac{x^2}{2}$, so:

${\int }_0^{2}xdx={\left[\frac{x^2}{2}\right]}_0^2=\frac{2^2}{2}-\frac{0^2}{2}=\frac{4}{2}=2$ 

Step 2: Evaluate the second integral ${\int }_2^4(4-x)dx$.

The antiderivative of 4 − x is $4x-\frac{x^2}{2}$, so:

${\int }_2^4(4-x)dx={\left[4x-\frac{x^2}{2}\right]}_2^4$ 

Now, evaluate at the limits:

$\begin{array}{rl}& =\left(4(4)-\frac{4^2}{2}\right)-\left(4(2)-\frac{2^2}{2}\right)\\ & =\left(16-\frac{16}{2}\right)-\left(8-\frac{4}{2}\right)=(16-8)-(8-2)\\ & =8-6=2\end{array}$

Step 3: Add the results.

Now, add the results from both integrals:

${\int }_0^{4}f(x)dx=2+2=4$

Thus, the value of the Riemann integral is 4 

Example 5: Evaluate the following integral ${\int }_0^2\left(3x^2+2x-1\right)dx$

Solution:

We need to find the antiderivative of the integrand $3x^2+2x-1$. Let's break it into parts.

Step 1: Find the antiderivative.

The antiderivative of  $3x^2\text{ is }{x}^3$, the antiderivative of 2x is x², and the antiderivative of −1 is −x.

Thus, the antiderivative of $3x^2+2x-1$ is:

$F(x)=x^3+x^2-x$

Step 2: Apply the principles of the Fundamental Theorem of Calculus.

Now, we apply the limits from 0 to 2:

${\int }_0^2\left(3x^2+2x-1\right)dx={\left[x^3+x^2-x\right]}_0^2$

Evaluating at the limits:

$\begin{array}{rl}& =\left(2^3+2^2-2\right)-\left(0^3+0^2-0\right)\\ & =(8+4-2)-(0)=10=10\end{array}$

Thus, the value of the integral is 10. 

Example 6: Evaluate the integral ${\int }_1^3(x-2)dx$

Solution:

The function f(x) = x - 2 is a simple linear function, so we can directly find its antiderivative.

Step 1: Find the antiderivative.

The antiderivative of x is $\frac{x^2}{2}$, and the antiderivative of −2 is −2x.

Thus, the antiderivative of x−2 is:

$F(x)=\frac{x^2}{2}-2x$

Step 2: Apply the principles of the Fundamental Theorem of Calculus.

Now, we evaluate from 1 to 3:

${\int }_1^3(x-2)dx={\left[\frac{x^2}{2}-2x\right]}_1^3$

Evaluating at the limits:

$\begin{array}{rl}& =\left(\frac{3^2}{2}-2(3)\right)-\left(\frac{1^2}{2}-2(1)\right)\\ & =\left(\frac{9}{2}-6\right)-\left(\frac{1}{2}-2\right)\\ & =\left(\frac{9}{2}-\frac{12}{2}\right)-\left(\frac{1}{2}-\frac{4}{2}\right)\\ & =\left(\frac{-3}{2}\right)-\left(\frac{-3}{2}\right)\\ & =-\frac{3}{2}+\frac{3}{2}=0\end{array}$

Thus, the value of the integral is 0 

Example 7: Evaluate the following integral ${\int }_0^{\pi }\sin (x)dx$

Solution:

We want to find the integral of $\sin (x)$ from 0 to π.

Step 1: Find the antiderivative.

The antiderivative of $\sin (x)is-\cos (x)$.

Step 2: Apply the principles of the Fundamental Theorem of Calculus.

Now, apply the limits from 0 to π:

${\int }_0^{\pi }\sin (x)dx=[-\cos (x){]}_0^{\pi }$ 

Evaluating at the limits:

$-\cos (\pi )+\cos (0)=-(-1)+1=1+1=2$

Thus, the value of the integral is 2.

Example 8: Evaluate the integral ${\int }_0^1\left(x^3+2x^2-3x+1\right)dx$

Solution:

We need to find the antiderivative of $x^3+2x^2-3x+1$.

Step 1: Find the antiderivative.

The antiderivative of x³ is $\frac{x^4}{4}$, the antiderivative of $2x^2\text{ is }\frac{2x^3}{3}$, the antiderivative of −3x is $\frac{3x^2}{2}$, and the antiderivative of 1 is x.

Thus, the antiderivative of $x^3+2x^2-3x+1$ is:

$F(x)=\frac{x^4}{4}+\frac{2x^3}{3}-\frac{3x^2}{2}+x$

Step 2: Apply the principles of the Fundamental Theorem of Calculus.

Now, apply the limits from 0 to 1:

${\int }_0^1\left(x^3+2x^2-3x+1\right)dx={\left[\frac{x^4}{4}+\frac{2x^3}{3}-\frac{3x^2}{2}+x\right]}_0^1$

Evaluating at the limits:

$=\left(\frac{1^4}{4}+\frac{2\left(1^3\right)}{3}-\frac{3\left(1^2\right)}{2}+1\right)-\left(\frac{0^4}{4}+\frac{2\left(0^3\right)}{3}-\frac{3\left(0^2\right)}{2}+0\right)=\left(\frac{1}{4}+\frac{2}{3}-\frac{3}{2}+1\right)$ 

Now, we simplify:

$=\frac{1}{4}+\frac{2}{3}-\frac{3}{2}+1$

The common denominator is 12:

$\begin{array}{rl}& =\frac{3}{12}+\frac{8}{12}-\frac{18}{12}+\frac{12}{12}\\ & =\frac{3+8-18+12}{12}=\frac{5}{12}\end{array}$

Thus, the value of the integral is:

$\frac{5}{12}$ 

7.0Practice Questions on Riemann Integral 

1. Evaluate: ${\int }_0^3\left(3x^2+2x\right)dx$
2. Show that $f(x)=\sin x$ is Riemann integrable on [0, π]
3. Compute: ${\int }_1^e\frac{1}{x}dx$
4. Is the function f(x) = 1 if x is rational, 0 otherwise, Riemann integrable on [0, 1]? Why or why not?