Definite Integration & Properties

Definite integration computes the net signed area under a curve f(x) between limits x = a and x = b. The Fundamental Theorem of Calculus (FTC) connects differentiation and integration: if F'(x) = f(x), then integral from a to b of f(x) dx = F(b) - F(a). This eliminates the constant of integration and gives a numerical value.

Properties of Definite Integrals:

Property 1 (Limits interchange): integral(a to b) f(x) dx = -integral(b to a) f(x) dx

Property 2 (Splitting): integral(a to b) f(x) dx = integral(a to c) f(x) dx + integral(c to b) f(x) dx for any c

Property 3 (Dummy variable): integral(a to b) f(x) dx = integral(a to b) f(t) dt. The variable of integration is a dummy.

Property 4 (King's Rule): integral(a to b) f(x) dx = integral(a to b) f(a+b-x) dx. This is the most powerful and frequently tested property. It works by substituting x → a+b-x.

Property 5 (Queen's Rule for [0,2a]): integral(0 to 2a) f(x) dx = integral(0 to a) [f(x) + f(2a-x)] dx. Special case: if f(2a-x) = f(x), integral = 2*integral(0 to a) f(x) dx. If f(2a-x) = -f(x), integral = 0.

Property 6 (Even/Odd on [-a,a]): integral(-a to a) f(x) dx = 2*integral(0 to a) f(x) dx if f is even, and = 0 if f is odd.

Property 7 (Periodicity): If f(x+T) = f(x), then integral(0 to nT) f(x) dx = n*integral(0 to T) f(x) dx.

Leibniz Rule for Differentiation Under the Integral Sign: d/dx [integral from g(x) to h(x) of f(t) dt] = f(h(x))*h'(x) - f(g(x))*g'(x)

Definite Integral as Limit of Sum (Riemann Sum): integral(0 to 1) f(x) dx = lim(n→infinity) (1/n) * sum(r=0 to n-1) f(r/n)

Steps: (1) Replace summation variable r/n by x, (2) Replace 1/n by dx, (3) Replace limits: r=0 gives x=0, r=n-1 gives x=1 (or r=n gives x=1). This converts summation problems to definite integrals.

Estimation and Inequalities:

• If f(x) >= g(x) on [a,b], then integral(a to b) f(x) dx >= integral(a to b) g(x) dx
• If m <= f(x) <= M on [a,b], then m(b-a) <= integral(a to b) f(x) dx <= M(b-a)
• Triangle inequality: |integral(a to b) f(x) dx| <= integral(a to b) |f(x)| dx

Wallis' Formula: integral(0 to $\frac{\pi}{2}$) sin^n(x) dx = integral(0 to $\frac{\pi}{2}$) cos^n(x) dx = [(n-1)(n-3)...]/[n(n-2)...] * ($\frac{\pi}{2}$ or 1) Factor of $\frac{\pi}{2}$ when n is even; factor of 1 when n is odd.

The key problem-solving concept is choosing the right property (especially King's Rule) to simplify the integral before attempting direct evaluation.