Integration Basics
Learn about antiderivatives and the area under curves. Master the Fundamental Theorem of Calculus and understand how integration reverses differentiation.
Problem Statement
Given a function f(x) = x², find the area under the curve from x = 0 tox = b using integration.
Example:
Input: f(x) = x², bounds [0, 2]
Output: ∫₀² x² dx = 8/3 ≈ 2.667
Method: F(x) = x³/3, so F(2) - F(0) = 8/3
Intuition
Integration calculates the area under a curve. While derivatives measure rates of change, integrals measure accumulation. The definite integral ∫ₐᵇ f(x) dx represents the total area between the function f(x) and the x-axis from a to b.
Key Concept: The Fundamental Theorem of Calculus connects differentiation and integration. If F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) - F(a). This means we can find areas by finding antiderivatives!
For f(x) = x², the antiderivative is F(x) = x³/3 (since d/dx[x³/3] = x²). To find the area from 0 to 2, we evaluate F(2) - F(0) = 8/3 - 0 = 8/3 ≈ 2.667.
Riemann Sums:
We can approximate the area using rectangles (Riemann sums). As we use more rectangles, the approximation gets closer to the exact integral value.
Interactive Visualization
Drag the upper bound to see how the area under the curve changes. Notice the Riemann rectangles approximating the integral.
Try it: Drag the red point to change the upper bound. The shaded area represents the definite integral ∫₀^b x² dx = [x³/3]₀^b = b³/3. Notice how the area grows as you increase b.
Implementation
import numpy as np
import matplotlib.pyplot as plt
from scipy import integrate
def numerical_integration(f, a, b, n=1000):
"""
Calculate definite integral using Riemann sum (midpoint rule)
∫ₐᵇ f(x) dx ≈ Σ f(xᵢ) · Δx
Args:
f: Function to integrate
a: Lower bound
b: Upper bound
n: Number of subdivisions
Returns:
Approximate value of the integral
"""
dx = (b - a) / n
total = 0
for i in range(n):
x_mid = a + (i + 0.5) * dx
total += f(x_mid) * dx
return total
# Example: ∫₀² x² dx
f = lambda x: x**2
a, b = 0, 2
# Numerical integration
result_numerical = numerical_integration(f, a, b)
print(f"Numerical: ∫₀² x² dx ≈ {result_numerical:.4f}")
# Analytical solution: F(x) = x³/3
# ∫₀² x² dx = [x³/3]₀² = 8/3 - 0 = 2.6667
F = lambda x: x**3 / 3
result_analytical = F(b) - F(a)
print(f"Analytical: ∫₀² x² dx = {result_analytical:.4f}")
# Using scipy for verification
result_scipy, error = integrate.quad(f, a, b)
print(f"SciPy: {result_scipy:.4f}")
# Visualize
x = np.linspace(0, 2, 100)
y = f(x)
plt.fill_between(x, 0, y, alpha=0.3, label=f'Area = {result_analytical:.4f}')
plt.plot(x, y, 'b-', linewidth=2, label='f(x) = x²')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Integration: Area Under Curve')
plt.legend()
plt.grid(True)
plt.show()Key Properties
Linearity
∫[af(x) + bg(x)]dx = a∫f(x)dx + b∫g(x)dx
Constants and sums can be separated
Reverses Differentiation
d/dx[∫f(x)dx] = f(x)
Integration undoes differentiation
Additivity
∫ₐᶜf(x)dx = ∫ₐᵇf(x)dx + ∫ᵇᶜf(x)dx
Areas can be split and combined
Zero Width
∫ₐᵃf(x)dx = 0
No area when bounds are equal
Related Topics
Derivatives
The inverse operation of integration
Substitution RuleComing Soon
Integration technique for composite functions
Integration by PartsComing Soon
Integration technique for products
ApplicationsComing Soon
Volume, work, and physics applications