Definite Integration

A definite integral represents the area under a curve between two points, called limits of integration. It is defined as the limit of a sum of the areas of rectangles.

Here, [a, b] is the interval, Δx = (b-a)/n is the width of each rectangle, and xᵢ is a point in the i-th subinterval. This method directly calculates the area by summing up infinitely many infinitesimally thin rectangles.

The Fundamental Theorem of Calculus provides a much more direct way to evaluate definite integrals. It connects differentiation and integration.

Part 1 (First fundamental theorem of integral calculus):

Let f be a continuous function on the closed interval [a, b] and let A(x) be the area function. Then A'(x) = f(x) for all x in [a, b].

Part 2 (Second fundamental theorem of integral calculus):

Let f be a continuous function on [a,b] and F be an anti-derivative of f. Then:

This means we can find the definite integral by finding the anti-derivative, evaluating it at the upper and lower limits, and subtracting.

To evaluate definite integrals using substitution, we change the variable of integration and also change the limits of integration to the new variable.

• P₀: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt
• P₁: ∫ₐᵇ f(x) dx = - ∫ₑᵃ f(x) dx. In particular, ∫ₐᵃ f(x) dx = 0
• P₂: ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx
• P₃: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b - x) dx
• P₄: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a - x) dx
• P₅: ∫₀²ᵃ f(x) dx = ∫₀ᵃ f(x) dx + ∫₀ᵃ f(2a - x) dx
• P₆: ∫₀²ᵃ f(x) dx = 2∫₀ᵃ f(x) dx, if f(2a-x)=f(x) and 0 if f(2a-x)=-f(x)
• P₇: (i) ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx, if f is an even function. (ii) ∫₋ₐᵃ f(x) dx = 0, if f is an odd function.

e.g. f(x) = cos(x) is an even function as cos(-x) = cos(x), f(x) = sin(x) is an odd function as sin(-x) = - sin(x), However f(x) = eˣ is neither odd nor even, so this property does not hold.

• P₈: ∫₀ⁿᵀ f(x) dx = n∫₀ᵀ f(x) dx, where f(x+T) = f(x)

e.g. T = 2π for f(x) = sin(x) and T = π for f(x) = |sin(x)|. However T does not exist for aperiodic functions e.g. for f(x) = eˣ .

For a periodic function like |sin(x)| with period T=π, the area under each arch is the same. Therefore, the integral from 0 to 2π is twice the integral from 0 to π.

• P₉: M(b - a) >= ∫ₐᵇ f(x) dx >= m(b - a)

where M is the maximum value and m is the minimum value of f(x) in the interval [a,b].

The area under the curve represented by the integral ∫ f(x) dx a to b shaded in blue is greater than the area of the inner rectangle (m(b-a)) shaded in green and less than the area of the outer rectangle (M(b-a)) shaded with red dots.

The definite integral of a function over an interval [a, b] gives the signed area between the function's graph and the x-axis. This is one of the most direct applications of integration.

Area Bounded by a Curve and the x-axis

The area of the region bounded by the curve y = f(x), the x-axis and the lines x = a and x = b is given by ∫ₐᵇ |f(x)| dx. If f(x) ≥ 0 for all x in [a, b], the area is simply ∫ₐᵇ f(x) dx.

Area Between Two Curves

The area of the region enclosed between two curves y = f(x) and y = g(x) from x = a to x = b, where f(x) ≥ g(x) for all x in [a, b], is given by:

This formula represents summing the heights of infinitesimally thin vertical strips between the curves.