Definite Integrals and the Fundamental Theorem of Calculus

🎯 Learning Objectives

1. Define the definite integral as the limit of Riemann sums
2. State the Fundamental Theorem of Calculus (Parts 1 and 2)
3. Evaluate definite integrals using the FTC
4. Compute the area between curves
5. Apply properties of definite integrals

📋 Prerequisites

• Indefinite Integrals — anti-derivatives, basic integration

📖 Core Content

16.1 Intuition: Accumulation of Area

💡 Why this matters: Definite integrals compute accumulated quantities: total distance from velocity, total charge from current, total probability from density, expected value in statistics.

16.2 Riemann Sums

16.3 Fundamental Theorem of Calculus (FTC)

16.4 Properties of Definite Integrals

16.5 Area Between Curves

16.6 Worked Examples

📐 Key Formulas — Summary Table

⚠️ Common Pitfalls

Pitfall 1: Forgetting to Subtract $F(a)$

Pitfall 2: Confusing Definite and Indefinite

Pitfall 3: Area For Absolute Value

📝 Practice Questions

Q1: Evaluate $\int_1^3 2x\,dx$.

$[x^2]_1^3 = 9 - 1 = 8$

$\boxed{8}$ Q2: Evaluate $\int_0^{\pi} \sin x\,dx$.

$[-\cos x]_0^{\pi} = -(-1) - (-1) = 1 + 1 = 2$

$\boxed{2}$ Q3: Find area under $y = e^x$ from $x = 0$ to $x = 1$.

$\int_0^1 e^x\,dx = [e^x]_0^1 = e - 1$

$\boxed{e - 1}$ Q4: Find the area between $y = x^2$ and $y = x + 2$.

Intersection: $x^2 = x + 2 \to x^2 - x - 2 = 0 \to (x-2)(x+1) = 0 \to x = -1, 2$ $\int_{-1}^2 [(x+2) - x^2]\,dx = \left[\frac{x^2}{2} + 2x - \frac{x^3}{3}\right]_{-1}^2 = \frac{9}{2}$

$\boxed{4.5}$ Q5: Evaluate $\int_0^2 (x^3 - 2x)\,dx$.

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

$\boxed{0}$ Q6: Find $\int_{-1}^1 |x|\,dx$.

$2\int_0^1 x\,dx = 2\left[\frac{x^2}{2}\right]_0^1 = 1$

$\boxed{1}$ Q7: Evaluate $\int_1^4 \frac{1}{\sqrt{x}}\,dx$.

$\int_1^4 x^{-1/2}\,dx = [2\sqrt{x}]_1^4 = 2(2) - 2(1) = 2$

$\boxed{2}$ Q8: Find $\int_0^1 \frac{1}{1+x^2}\,dx$.

$[\arctan x]_0^1 = \frac{\pi}{4} - 0 = \frac{\pi}{4}$

$\boxed{\frac{\pi}{4}}$ Q9: Average value of $f(x) = x^2$ on $[0, 3]$.

$\frac{1}{3}\int_0^3 x^2\,dx = \frac{1}{3}\left[\frac{x^3}{3}\right]_0^3 = \frac{1}{3} \cdot 9 = 3$

$\boxed{3}$ Q10: Find $\int_0^1 e^{2x}\,dx$.

$\left[\frac{e^{2x}}{2}\right]_0^1 = \frac{e^2}{2} - \frac{1}{2} = \frac{e^2 - 1}{2}$

$\boxed{\frac{e^2 - 1}{2}}$

🔗 Cross-References

• Previous: Indefinite Integrals
• Next: Integration Techniques
• Across courses: BSMA1002 Stats 1 (probability density functions → areas under curves) Join Discord Previous9.1 Indefinite IntegralsNext9.3 Integration Techniques