Definition of Integral

Riemann Sum Sum given by the formula

Integral If f is defined on the interval [ a, b] and the limit exists, then this limit is called the definite integral of f from a to b

f(x) is called the integrand a is the lower limit b is the upper limit

Example Express the Riemann sum below as an integral on the interval [ 0, π ]

Example Evaluate the integral below using the limit method 0 3 𝑥 2 𝑑𝑥

Homework Evaluate using the limit method 1) 0 3 5𝑑𝑥 2) 0 2 2𝑥𝑑𝑥 3) 0 1 3 𝑥 2 𝑑𝑥

Areas Using Geometry Using the region corresponding to each definite integral, evaluate 1) 2) 3)

Properties of Integrals

Example 1) 2)

Example If it is known the and find