Definite integrals extend indefinite integration by adding limits of integration and computing a numerical result rather than a function.
## Core concept
A definite integral is written as $\int_a^b f(x) \, dx$ where: - $a$ = lower limit - $b$ = upper limit - $f(x)$ = integrand - The result is a number, not a function
Fundamental Theorem of Calculus (FTC): $$\int_a^b f(x) \, dx = [F(x)]_a^b = F(b) - F(a)$$
where $F(x)$ is any antiderivative of $f(x)$.
This means: find the indefinite integral first, then evaluate at the upper limit and subtract the value at the lower limit.
## Formula / rule
Basic evaluation steps: 1. Find the antiderivative $F(x)$ using standard integral formulas 2. Apply the upper limit: $F(b)$ 3. Apply the lower limit: $F(a)$ 4. Subtract: $F(b) - F(a)$
Key properties of definite integrals: - $\int_a^a f(x) \, dx = 0$ (same limits) - $\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$ (reverse limits change sign) - $\int_a^c f(x) \, dx = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx$ (addition of intervals) - $\int_a^b [f(x) + g(x)] \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx$ (linearity) - $\int_a^b k \cdot f(x) \, dx = k \int_a^b f(x) \, dx$ (constant multiple)
Standard definite integral formulas: - $\int_a^b x^n \, dx = \left[\frac{x^{n+1}}{n+1}\right]_a^b$ (power rule; $n \neq -1$) - $\int_a^b e^x \, dx = [e^x]_a^b$ - $\int_a^b \frac{1}{x} \, dx = [\ln|x|]_a^b$ (for $x > 0$) - $\int_a^b \sin x \, dx = [-\cos x]_a^b$ - $\int_a^b \cos x \, dx = [\sin x]_a^b$
## Worked example
Evaluate: $\int_1^3 (2x^2 + 3) \, dx$
Solution: 1. Find antiderivative: $\int (2x^2 + 3) \, dx = \frac{2x^3}{3} + 3x + C$ 2. Apply limits: $\left[\frac{2x^3}{3} + 3x\right]_1^3$ 3. Upper limit ($x = 3$): $\frac{2(27)}{3} + 3(3) = 18 + 9 = 27$ 4. Lower limit ($x = 1$): $\frac{2(1)}{3} + 3(1) = \frac{2}{3} + 3 = \frac{11}{3}$ 5. Result: $27 - \frac{11}{3} = \frac{81 - 11}{3} = \frac{70}{3} \approx 23.33$
## Common exam applications
- Area under a curve: $\int_a^b f(x) \, dx$ gives the area between $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$
- Marginal analysis in business: Computing total cost, revenue, or profit from marginal functions
- Accumulation problems: Total distance from velocity, total consumption from rate of consumption
- Average value of a function: $\text{Average} = \frac{1}{b-a} \int_a^b f(x) \, dx$
## Common mistakes
- Forgetting to subtract: Writing only $F(b)$ instead of $F(b) - F(a)$
- Sign errors on reversal: Reversing limits but forgetting the negative sign
- Ignoring the constant: Adding a constant $C$ to definite integral (constant cancels: $F(b) + C - F(a) - C = F(b) - F(a)$)
- Algebraic slips at boundaries: Miscalculating $F(b)$ or $F(a)$; always substitute carefully
- Mismatched integral tables: Using an indefinite integral formula incorrectly