Integrals & Fundamental Theorem

🔢 Integrals & The Fundamental Theorem of Calculus

Integration is the process of finding the accumulation of a quantity, such as the area under a curve or the total distance traveled.

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🟢 1. Antiderivatives and Indefinite Integrals

An antiderivative of a function $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$.

The indefinite integral represents the family of all antiderivatives: $\int f(x) dx = F(x) + C$ where $C$ is the constant of integration.

Common Integrals

• Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$).
• Exponential: $\int e^x dx = e^x + C$.
• Trigonometric: $\int \cos(x) dx = \sin(x) + C$.
• Logarithmic: $\int \frac{1}{x} dx = \ln|x| + C$.

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🟡 2. Definite Integrals & Riemann Sums

The definite integral represents the area under the curve $f(x)$ between $x=a$ and $x=b$.

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

Riemann Sums

A Riemann sum is an approximation of the area under a curve by dividing it into rectangles.

• Left/Right Endpoints: Using the left or right side of each subinterval for the height.
• Midpoint Rule: Using the middle of each subinterval for more accuracy.
• Trapezoidal Rule: Using trapezoids instead of rectangles.

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🔴 3. The Fundamental Theorem of Calculus (FTC)

The FTC links differentiation and integration, showing that they are essentially inverse operations.

Part 1: Derivative of an Integral

If $g(x) = \int_a^x f(t) dt$, then $g'(x) = f(x)$.

Part 2: Evaluation of a Definite Integral

If $F$ is an antiderivative of $f$ on $[a, b]$, then: $\int_a^b f(x) dx = F(b) - F(a)$

Tip

This theorem is why we find antiderivatives to solve area problems!

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🎯 4. Integration Techniques

1. Integration by Substitution (U-Substitution)

Reverses the Chain Rule: $\int f(g(x)) g'(x) dx = \int f(u) du$

2. Integration by Parts

Reverses the Product Rule: $\int u dv = uv - \int v du$

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💡 Practical Example: Numerical Integration (Simpson’s Rule)

When we can’t find an antiderivative analytically, we use numerical methods.

import numpy as np

def f(x):
    return x**2

def simpsons_rule(f, a, b, n):
    if n % 2: n += 1 # n must be even
    h = (b - a) / n
    x = np.linspace(a, b, n + 1)
    y = f(x)
    S = h/3 * np.sum(y[0:-1:2] + 4*y[1::2] + y[2::2])
    return S

# Example: Integral of x^2 from 0 to 1 (should be 1/3)
area = simpsons_rule(f, 0, 1, 100)
print(f"Approximated area: {area}")

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🚀 Key Takeaways

• Integration sums up small parts to find a total.
• The FTC connects derivatives and integrals.
• U-substitution and Integration by Parts are the primary techniques for solving integrals.