Fundamental Theorem of Calculus Indefinite Integrals 4.3- 4.4 Fundamental Theorem of Calculus Indefinite Integrals

Example: Evaluate A(x) Area between the graph of f(x) and the x-axis over the interval [2,x]

Using geometry:

Using integration:

Fundamental Theorem of Calculus

Fundamental Theorem of Calculus – part 1: Fundamental Theorem of Calculus – simplest form: Fundamental Theorem of Calculus – more general form:

Fundamental Theorem of Calculus - part 2: Suppose that f is bounded on the interval [a,b], and that F is an antiderivative of f, i.e.,  F’ = f.     Then:

Example 1: Solution:

Example 2: Solution:

Example 3: Solution:

More practice problems with solutions: http://tutorial.math.lamar.edu/Classes/CalcI/ComputingDefiniteIntegrals.aspx

4.5 Substitution Rule

Example 1: Find  x3 cos(x4 + 2) dx. Solution: We make the substitution u = x4 + 2 because its differential is du = 4x3 dx, which, apart from the constant factor 4, occurs in the integral. Thus, using x3 dx = du and the Substitution Rule, we have  x3 cos(x4 + 2) dx =  cos u  du =  cos u du

Example 1 – Solution cont’d = sin u + C = sin(x4 + 2) + C Notice that at the final stage we had to return to the original variable x.

Example 2: Evaluate . Solution: Let u = 2x + 1. Then du = 2 dx, so dx = du. So: 4