#C04 · Calculus & Analysis · Lv.3View in 3D Mind-Map

Fundamental Theorem of Calculus (FTC)

Newton, Leibniz & Isaac Barrow · 1669 CE

\int_a^b f(x)\,dx = F(b) - F(a) \quad (F'(x) = f(x))

\frac{d}{dx} \int_a^x f(t)\,dt = f(x)

Unites differentiation and integration as inverse operations, allowing the area under any curve to be computed simply from antiderivative endpoints F(b) - F(a).

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Lower limit a

Upper limit b

\int_{0}^{3} x^2\,dx = \left[\frac{x^3}{3}\right]_{0}^{3} = \frac{27.0 - 0.0}{3} = 9.0000

Definite Integral

9.0000

Antiderivative ΔF

9.0000

Mean Value

3.000

Prerequisite Path (9 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 2#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 3#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 4#A06Logarithm Laws & Change of Base

\log_a(xy) = \log_a x + \log_a y

STEP 5#A08Binomial Theorem & Pascal's Triangle

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

STEP 6#C01Limits, Continuity & Euler's Number e

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828

STEP 7#A07Arithmetic & Geometric Series Sum

S_n = \frac{a(1 - r^n)}{1 - r}

STEP 8#C02Derivative Definition & Instantaneous Rate

f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

STEP 9#C04Fundamental Theorem of Calculus (FTC)

\int_a^b f(x)\

🚀 Next Unlocks (Derived Formulas) (4)

#C05Integration by Substitution & by Parts #C06Taylor & Maclaurin Series Expansion #C09Fourier Series & Fourier Transform (FFT)#P06Normal (Gaussian) Distribution & Z-Score

Geometric & Intuitive Insight

시속 f(t) km로 달린 자동차의 속도 그래프 아래 넓이(이동거리)는, 결국 출발할 때 계기판 누적거리 F(a)와 도착할 때 계기판 누적거리 F(b)의 차이 F(b) - F(a)와 정확히 같습니다.

Step-by-Step Derivation

Step 1.1단계: 넓이 함수 A(x) = ∫_a^x f(t) dt 에 대해 증분 ΔA = A(x+h) - A(x) ≈ f(x)·h 입니다.
Step 2.2단계: 양변을 h로 나누고 h → 0 극한을 취하면 A'(x) = f(x) 가 되어 넓이 함수의 도함수가 원래 함수 f(x)임이 증명됩니다.
Step 3.3단계: 따라서 임의의 부정적분 F(x)에 대해 ∫_a^b f(x) dx = F(b) - F(a) 가 성립합니다.

AI & Coding Real-World Use

Physics engine velocity-to-position integration, 3D ray marching rendering equation, and probability CDF

#Physics RK4 Integrator#3D Rendering Equation#Probability CDF

const simpson = (f:(x:number)=>number, a:number, b:number) => ((b-a)/6)*(f(a)+4*f((a+b)/2)+f(b));

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