Symbolic Integral Notation Lesson 5.1

Why? Why is the area of the yellow rectangle at the end = a b

Review a b We partition the interval into n sub-intervals f(x)

The Definite Integral So the definition says that the definite integral of an integrable function can be approximated to within any desired degree of accuracy by a Riemann sum. We know that if f happens to be positive, then the Riemann sum can be interpreted as a sum of areas of approximating rectangles (see Figure 1). If f (x)  0, the Riemann sum f (xi*) x is the sum of areas of rectangles. Figure 1

Review Evaluate f(x) of kth sub-interval for k = 1, 2, 3, … n Write an expression for x1, x2, and x3. x1 = a + 1( ) x2 = a + 2( ) x3 = a + 3( ) xn = a + n( ) What is the expression pattern? a b f(x) x1 x2 x3

Area bounded by the curve, x-axis, x=a and x=b

Area bounded by the curve, x-axis, x=a and x=b

Review What is the area expression for each sub - interval? f(x) a b k1 k2 k3

Review a b What is the sum of the area from a to b? f(x) k1 k2 k3

Riemann Sum Form the sum This is the Riemann sum associated with the function f the given partition P the chosen subinterval representatives

The Definite Integral When Leibniz chose the notation for an integral, he chose the ingredients as reminders of the limiting process. In general, when we write we replace “lim ” by , “ .” by x, and “x” by dx.

Example 1 Express as an integral on the interval [0, ]. Solution:

The Definite Integral The definite integral is the limit of the Riemann sum We say that f is integrable when the number I can be approximated as accurate as needed by making ||P|| sufficiently small f must exist on [a,b] and the Riemann sum must exist