Definite Integrals: Quick Revision

A brief note on Definite Integrals. Read it before a test. Try each question at the end before you open its answer.

The Fundamental Theorem

• Signed area: $\int_a^b f(x)\,dx$ is positive above the axis and negative below it.
• Part 1: if $A(x) = \int_a^x f(t)\,dt$ and $f$ is continuous, then $A'(x) = f(x)$.
• Part 2: $\int_a^b f(x)\,dx = F(b) - F(a)$ for any antiderivative $F$.
• Condition: $f$ must be continuous on $[a, b]$.

Substitution and properties

• New limits: with $u = g(x)$, the limits become $g(a)$ and $g(b)$.
• Split: $\int_a^b f = \int_a^c f + \int_c^b f$. Split at a sign change for $|f|$.
• Swap: $\int_a^b f = -\int_b^a f$.
• Even and odd: $\int_{-a}^{a} f = 2\int_0^a f$ for even $f$, and $0$ for odd $f$.
• Reflection: $\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$.

The limit of a sum

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

A worked example

Find $\displaystyle\int_1^3 \left(3x^2 + 2\right)dx$.

1. Antiderivative. $F(x) = x^3 + 2x$.
2. Evaluate. $F(3) - F(1) = (27 + 6) - (1 + 2) = 30$.

Practice with new numbers

A program to try

The program finds the area of the worked example by adding thin strips, and compares it with $F(3) - F(1)$.

Mistakes to avoid

• Forgetting $F(a)$. The answer is $F(b) - F(a)$.
• Old limits after a substitution. Change the limits with the variable.
• Area and signed area. Split where $f$ changes sign.
• Wrong width in a sum. $h = \dfrac{b - a}{n}$.

Check yourself

1. Find $\displaystyle\int_0^2 \left(x + 1\right)dx$.

2. Find $\displaystyle\int_{-3}^{3} x^3\,dx$.

3. If $A(x) = \displaystyle\int_0^x \sin t\,dt$, what is $A'(x)$?

4. What happens to the limits of $\displaystyle\int_0^1 x\,e^{x^2}\,dx$ when $u = x^2$?

5. What is the width $h$ when $[1, 4]$ is cut into $n$ equal strips?

Sources

• OpenStax, Calculus Volume 1, Chapter 5, sections 5.2, 5.3 and 5.5.
• NCERT, Mathematics Part II for Class XII, Chapter 7, the sections on the definite integral, the Fundamental Theorem of Calculus and some properties of definite integrals.