Section 5.2 Definite Integrals

Definite Integral and Area All continuous functions are integrable. If y = f(x) is nonnegative and integrable over a closed interval [a, b], then the area under the curve y = f(x) from a to b is the integral of f from a to b. A = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 (It doesn’t matter which letter you use to denote the variable of integration).

The Integral Sign All together, this statement is read “the integral from a to b of f of x dx.”

Negative Area If f(x) < 0, then Area = - 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 For any integrable function, 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = area above the x-axis – area below the x-axis. Use approximation methods to find 0 6 (𝑥 −3) 2 − 3 𝑑𝑥

Integrals on the calculator We have already discussed finding an integral using Riemann sums and using geometric methods. To calculate a definite integral on the calculator, use the fnInt command found in the “math” menu. Use this command to calculate integrals that you estimated using Riemann sums.

Constant Function If f(x) = c, where c is a constant on the interval [a,b], then 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = ? = 𝑎 𝑏 𝑐 𝑑𝑥 = c (b – a)