Mathematics I — Week 9: Integration, Areas & the Fundamental Theorem

TL;DR

How fast is something changing?

How much change has accumulated?

$$\boxed{\text{Indefinite integral}=\text{antiderivative}}$$ $$\boxed{\text{Definite integral}=\text{signed accumulated area}}$$

---

1. Antiderivative intuition

$$F'(x)=f(x).$$ $$F(x)=x^2.$$ $$F'(x)=2x.$$ $$2x$$ $$x^2.$$

---

2. Indefinite integral

$$\boxed{ \int f(x)\,dx }$$ $$F'(x)=f(x),$$ $$\boxed{ \int f(x)\,dx=F(x)+C }$$ $$C$$

---

3. Why the constant $C$ appears

$$\frac{d}{dx}(x^2)=2x$$ $$\frac{d}{dx}(x^2+5)=2x$$ $$\frac{d}{dx}(x^2-100)=2x.$$ $$x^2+C$$ $$2x.$$ $$\int 2x\,dx=x^2+C.$$

---

4. Basic integration notation

$$\int f(x)\,dx,$$

• $\int$ = integral sign
• $f(x)$ = integrand
• $dx$ = variable of integration

---

5. Power rule for integration

$$n\neq -1,$$ $$\boxed{ \int x^n\,dx = \frac{x^{n+1}}{n+1}+C }$$ $$\frac{d}{dx}x^n=nx^{n-1}.$$

---

6. Examples

$$\int x^2\,dx = \frac{x^3}{3}+C$$ $$\frac{d}{dx}\left(\frac{x^3}{3}\right)=x^2.$$ $$\int x^5\,dx = \frac{x^6}{6}+C.$$

---

7. Integrating constants

$$\int c\,dx = cx+C.$$ $$\frac{d}{dx}(cx)=c.$$ $$\boxed{ \int 7\,dx=7x+C }$$

---

8. Constant multiple rule

$$\boxed{ \int c f(x)\,dx = c\int f(x)\,dx }$$ $$\int 5x^3\,dx = 5\cdot\frac{x^4}{4}+C$$ $$\boxed{ \frac54x^4+C }$$

---

9. Sum and difference rules

$$\boxed{ \int[f(x)+g(x)]\,dx = \int f(x)\,dx+\int g(x)\,dx }$$ $$\boxed{ \int[f(x)-g(x)]\,dx = \int f(x)\,dx-\int g(x)\,dx }$$

---

10. Polynomial integration

$$\int(3x^2-4x+5)\,dx.$$ $$=3\frac{x^3}{3} -4\frac{x^2}{2} +5x+C.$$ $$\boxed{ x^3-2x^2+5x+C }$$

---

11. Check your answer by differentiating

$$\int f(x)\,dx=F(x)+C,$$ $$F'(x).$$ $$F'(x)=f(x),$$

---

12. Special case: $\int \frac1x\,dx$

$$n=-1$$ $$n+1=0.$$ $$\boxed{ \int\frac1x\,dx = \ln|x|+C }$$ $$x\neq0.$$ $$\frac{d}{dx}\ln|x|=\frac1x$$

---

13. Exponential integrals

$$\frac{d}{dx}e^x=e^x,$$ $$\boxed{ \int e^x\,dx=e^x+C }$$ $$\boxed{ \int a^x\,dx = \frac{a^x}{\ln a}+C }$$ $$a>0,\quad a\neq1.$$

---

14. Integration as accumulated area

$$f(x)\geq0$$ $$[a,b].$$ $$y=f(x)$$ $$\boxed{ \int_a^b f(x)\,dx }$$

---

15. Definite integral notation

$$\int_a^b f(x)\,dx$$

• $a$ = lower limit
• $b$ = upper limit
• $f(x)$ = integrand

---

16. Why integration is related to summation

$$[a,b]$$ $$\text{area}\approx \text{height}\times\text{width}.$$ $$\sum f(x_i)\Delta x.$$ $$\Delta x\to0,$$ $$\boxed{ \int_a^b f(x)\,dx }$$

---

17. Riemann sum intuition

$$\sum_{i=1}^n f(x_i^*)\Delta x.$$ $$n\to\infty,$$ $$\Delta x\to0.$$ $$\boxed{ \int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^n f(x_i^*)\Delta x }$$

---

18. Fundamental Theorem of Calculus

$$F'(x)=f(x),$$ $$\boxed{ \int_a^b f(x)\,dx = F(b)-F(a) }$$

1. find an antiderivative
2. evaluate at upper bound
3. subtract value at lower bound

---

19. Example

$$\int_0^2 x\,dx.$$ $$F(x)=\frac{x^2}{2}.$$ $$\int_0^2x\,dx = \left[\frac{x^2}{2}\right]_0^2.$$ $$= \frac{2^2}{2} - \frac{0^2}{2}$$ $$=2.$$ $$\boxed{2}$$

---

20. Bracket notation

$$\left[F(x)\right]_a^b$$ $$F(b)-F(a).$$ $$\left[\frac{x^3}{3}\right]_1^4$$ $$\frac{4^3}{3}-\frac{1^3}{3}.$$

---

21. No $+C$ in a definite integral

$$\int_a^b f(x)\,dx,$$ $$+C.$$ $$F(x)+C.$$ $$[F(b)+C]-[F(a)+C]$$ $$F(b)-F(a).$$

---

22. Example: polynomial definite integral

$$\int_1^3(2x+1)\,dx.$$ $$x^2+x.$$ $$[x^2+x]_1^3.$$ $$9+3=12.$$ $$1+1=2.$$ $$\boxed{10}$$

---

23. Geometric verification

$$f(x)=2x+1$$ $$x=1$$ $$x=3,$$ $$f(1)=3$$ $$f(3)=7.$$ $$2.$$ $$\frac12(3+7)(2)=10.$$

---

24. Signed area

$$f(x)>0,$$ $$f(x)<0,$$

---

25. Example below the axis

$$f(x)=-2$$ $$[0,3].$$ $$\int_0^3 -2\,dx = -6.$$ $$6,$$ $$-6.$$

---

26. Net area vs total area

$$\int_a^b f(x)\,dx$$ $$\boxed{ \int_a^b |f(x)|\,dx }$$

---

27. Example: signed cancellation

$$f(x)=x$$ $$[-1,1].$$ $$\int_{-1}^{1}x\,dx=0.$$ $$-\frac12.$$ $$+\frac12.$$ $$1.$$

---

28. Symmetry shortcut: odd functions

$$f(-x)=-f(x),$$ $$[-a,a],$$ $$\boxed{ \int_{-a}^a f(x)\,dx=0 }$$ $$f(x)=x^3.$$ $$\int_{-2}^2x^3\,dx=0.$$

---

29. Symmetry shortcut: even functions

$$f(-x)=f(x),$$ $$\boxed{ \int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx }$$ $$f(x)=x^2.$$

---

30. Basic properties of definite integrals

Same limits

$$\boxed{ \int_a^a f(x)\,dx=0 }$$

---

Reversing limits

$$\boxed{ \int_a^b f(x)\,dx = -\int_b^a f(x)\,dx }$$

---

Splitting interval

$$\boxed{ \int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx }$$

---

31. Linearity of definite integrals

$$\boxed{ \int_a^b [\alpha f(x)+\beta g(x)]\,dx = \alpha\int_a^b f(x)\,dx + \beta\int_a^b g(x)\,dx }$$

---

32. Area between two curves

$$f(x)\geq g(x)$$ $$[a,b].$$ $$\boxed{ \int_a^b [f(x)-g(x)]\,dx }$$ $$\text{top} - \text{bottom}.$$

---

33. Example: area between curves

$$y=x$$ $$y=x^2$$ $$x=0$$ $$x=1.$$ $$x\geq x^2.$$ $$\int_0^1(x-x^2)\,dx.$$ $$\left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_0^1.$$ $$\frac12-\frac13 = \boxed{\frac16}.$$

---

34. Find intersection points first

$$f(x)=g(x)$$ $$x=x^2.$$ $$x^2-x=0$$ $$x(x-1)=0.$$ $$x=0,\quad1.$$

---

35. Antiderivative families

$$f'(x)=6x.$$ $$f(x)=3x^2+C.$$ $$f(1)=10,$$ $$3(1)^2+C=10.$$ $$C=7.$$ $$\boxed{ f(x)=3x^2+7 }$$

---

36. Recovering position from velocity

$$v(t)=s'(t).$$ $$t=a$$ $$t=b$$ $$\boxed{ \int_a^b v(t)\,dt = s(b)-s(a) }$$

---

37. Displacement vs distance

$$\int_a^b v(t)\,dt$$ $$\int_a^b |v(t)|\,dt.$$

---

38. Fundamental Theorem — deeper intuition

$$A(x)=\int_a^x f(t)\,dt.$$ $$\boxed{ A'(x)=f(x) }$$

the instantaneous rate at which accumulated area changes is the current height of the function.

---

39. Why does $A'(x)=f(x)$ make sense?

$$\Delta x.$$ $$\Delta A\approx f(x)\Delta x.$$ $$\frac{\Delta A}{\Delta x}\approx f(x).$$ $$\Delta x\to0,$$ $$A'(x)=f(x).$$

---

40. Substitution intuition

$$\int 2x(x^2+1)^3\,dx.$$ $$\frac{d}{dx}(x^2+1)=2x.$$ $$u=x^2+1.$$ $$du=2x\,dx.$$ $$\int u^3\,du.$$ $$\frac{u^4}{4}+C.$$ $$\boxed{ \frac{(x^2+1)^4}{4}+C }$$

---

41. Reverse chain-rule mental model

$$\frac{d}{dx}F(g(x)) = F'(g(x))g'(x).$$ $$\boxed{ \int F'(g(x))g'(x)\,dx = F(g(x))+C }$$

---

42. Example

$$\int 3x^2e^{x^3}\,dx.$$ $$u=x^3.$$ $$du=3x^2\,dx.$$ $$\int e^u\,du = e^u+C.$$ $$\boxed{ e^{x^3}+C }$$

---

43. Another example

$$\int\frac{2x}{x^2+1}\,dx.$$ $$u=x^2+1.$$ $$du=2x\,dx.$$ $$\int\frac1u\,du = \ln|u|+C.$$ $$x^2+1>0,$$ $$\boxed{ \ln(x^2+1)+C }$$

---

44. Definite integral with substitution

$$\int_0^1 2x(x^2+1)\,dx.$$ $$u=x^2+1.$$ $$x=0,$$ $$u=1.$$ $$x=1,$$ $$u=2.$$ $$\int_1^2u\,du.$$ $$\left[\frac{u^2}{2}\right]_1^2 = 2-\frac12 = \boxed{\frac32}.$$

---

45. Average value of a function

$$f(x)$$ $$[a,b]$$ $$\boxed{ f_{\text{avg}} = \frac1{b-a} \int_a^b f(x)\,dx }$$ $$\text{average} = \frac{\text{total}}{\text{amount}}.$$

---

46. Example: average value

$$f(x)=x$$ $$[0,2],$$ $$f_{\text{avg}} = \frac12 \int_0^2x\,dx.$$ $$\int_0^2x\,dx=2.$$ $$\boxed{ f_{\text{avg}}=1 }$$

---

$$\int f(x)\,dx,$$ $$+C.$$

---

$$\int x^3\,dx=3x^2.$$ $$\boxed{ \int x^3\,dx=\frac{x^4}{4}+C }$$

add $1$ to exponent, then divide by new exponent.

---

$$\frac{x^{n+1}}{n+1}$$ $$n=-1.$$ $$\boxed{ \int x^{-1}\,dx = \ln|x|+C }$$

---

$$\int_a^b f(x)\,dx$$

---

$$\boxed{ F(b)-F(a) }$$ $$F(a)-F(b).$$

---

$$\int f(x)g(x)\,dx = \left(\int f\right) \left(\int g\right).$$

• substitution
• other methods later

---

$$\frac{d}{dx}f(x)$$ $$\int f(x)\,dx$$

---

54. Derivative ↔ integral table

---

55. Integration formula sheet

Power rule

$$\boxed{ \int x^n\,dx = \frac{x^{n+1}}{n+1}+C }$$

Constant

$$\boxed{ \int c\,dx=cx+C }$$

Reciprocal

$$\boxed{ \int\frac1x\,dx = \ln|x|+C }$$

Exponential

$$\boxed{ \int e^x\,dx=e^x+C }$$

General exponential

$$\boxed{ \int a^x\,dx = \frac{a^x}{\ln a}+C }$$

---

56. Definite integral formula sheet

Fundamental theorem

$$F'=f,$$ $$\boxed{ \int_a^b f(x)\,dx = F(b)-F(a) }$$

Reverse bounds

$$\boxed{ \int_a^b f = -\int_b^a f }$$

Split intervals

$$\boxed{ \int_a^b f = \int_a^c f+\int_c^b f }$$

Area between curves

$$\boxed{ A= \int_a^b (\text{top}-\text{bottom})\,dx }$$

---

57. Full worked example

$$\int_0^2(x^2+2x+1)\,dx.$$ $$F(x) = \frac{x^3}{3}+x^2+x.$$ $$\left[ \frac{x^3}{3}+x^2+x \right]_0^2.$$ $$x=2:$$ $$\frac83+4+2 = \frac83+6 = \frac{26}{3}.$$ $$x=0:$$ $$0.$$ $$\boxed{ \frac{26}{3} }$$

---

58. Full area example crossing the axis

$$y=x$$ $$x=-2$$ $$x=3.$$ $$x=0.$$ $$-\int_{-2}^{0}x\,dx + \int_0^3x\,dx.$$ $$\int_{-2}^{0}x\,dx = \left[\frac{x^2}{2}\right]_{-2}^0 = 0-2 = -2.$$ $$2.$$ $$\int_0^3x\,dx = \frac92.$$ $$2+\frac92 = \boxed{\frac{13}{2}}.$$

---

59. Weeks 7–9 as one calculus story

text
Week 7
LIMITS
"What happens as we approach?"
        ↓
Week 8
DERIVATIVE
"How fast is it changing now?"
        ↓
Week 9
INTEGRAL
"How much has accumulated?"

$$\boxed{ \text{differentiation and integration are inverse processes} }$$

---

60. What you should actually master

1. Explain what an antiderivative is.
2. Understand why indefinite integrals include $+C$.
3. Apply the integration power rule.
4. Integrate polynomials.
5. Handle: $$\int \frac1x\,dx.$$
6. Integrate basic exponentials.
7. Interpret definite integrals as signed accumulation.
8. Use: $$F(b)-F(a).$$
9. Distinguish signed area from geometric area.
10. Split integrals at zeros/intersections.
11. Compute area between two simple curves.
12. Understand the Fundamental Theorem of Calculus.
13. Recognize basic substitution as the reverse chain rule.
14. Recover a function from its derivative plus an initial condition.
15. Interpret integrals as displacement/accumulation.

---

Quick Recall & Flashcards

• Fundamental Theorem of Calculus (Part 1) :: $\frac{d}{dx} \left[\int_a^x f(t) dt\right] = f(x)$
• Fundamental Theorem of Calculus (Part 2) :: $\int_a^b f(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$
• Power rule for integration $\int x^n dx$ ($n \neq -1$) :: $\frac{x^{n+1}}{n+1} + C$
• Indefinite integral $\int \frac{1}{x} dx$ :: $\ln|x| + C$
• Indefinite integral $\int e^{kx} dx$ :: $\frac{1}{k} e^{kx} + C$
• Geometric meaning of $\int_a^b f(x) dx$ :: Net signed area between curve $y = f(x)$ and x-axis
• u-substitution is the integral counterpart of :: The Chain Rule in differentiation

---

Practice & Self-Check