Integrals: area under a curve

The integral of f from a to b is the area between the curve and the x-axis over that interval. It answers questions like "how far did the car go, given its speed at every moment?"

Thin rectangles

Cut the interval into n strips of width w = (b − a) ÷ n. In each strip, draw a rectangle as tall as the curve at the strip's midpoint, and add the areas. This is a Riemann sum.

def f(x):
    return x * x

a, b = 0, 3
for n in [2, 4, 8, 16]:
    width = (b - a) / n
    total = 0
    for i in range(n):
        total += f(a + (i + 0.5) * width) * width
    print(n, "strips:", round(total, 5))
print("exact: 3³/3 =", 3 ** 3 / 3)

For f(x) = x² from 0 to 3, the sums approach 9. Doubling the number of strips makes the midpoint error about four times smaller.

The fundamental theorem of calculus

Integration undoes differentiation. If F′ = f (F is an antiderivative of f), then:

the integral of f from a to b = F(b) − F(a)

For f(x) = x², F(x) = x³ ÷ 3, so the area is 27/3 − 0 = 9 exactly, with no rectangles at all.

Where it is used

Distance from speed, probability from a density curve (the normal curve's areas in the Probability course), work from force, and the total of any quantity that changes continuously.

Common mistakes

• Area below the x-axis counts as negative. The integral of sin x from 0 to 2π is 0, though the curve is not flat.
• Forgetting that an antiderivative is only defined up to a constant: x³/3 + 7 works too, and the 7 cancels in F(b) − F(a).

Mark lesson complete