4.1 Antiderivatives 1 Definition: The antiderivative of a function f is a function F such that F’=f. Note: Antiderivative is not unique! Example: Show that F 1 =x 3 +1 and F 2 =x 3 +2 are both antiderivatives of f =3x 2. Solution: Differentiate the functions: Invert the problem: given the derivative of a function, find the function itself. Notation: the derivative is denoted as f; the antiderivative as F so that F’=f.

Theorem: If F(x) is an atiderivative if a function f(x), then any function F(x)+C, where C is any constant, is also an antiderivative of the same function. Note: the constant C is called the constant of integration. Antiderivative of Power Function: Remember: differentiation does not act on a constant multiplier. Thus, antidifferentiation does not act on a constant multiplier either! Exercise: Find the antiderivative of, where k is a constant. 2

3 Another very important rule: Antiderivative of a sum is a sum of antiderivatives. (based on the analogous property of the derivative) Exercises: Find the antiderivatives of the following functions

4 Homework: Section 4.1: 1,3,5,7,9,11,13,15.

4.2 The Area Problem 5 Then, the area is approximated by the sum of areas of these rectangles: Agenda: calculate the area under a curve bounded by the x-axis, and vertical lines x=a and x=b. Approach: approximate this region by many vertical rectangles.

6 To make the approximation better, we need to increase the number of the rectangles (and simultaneously decrease their bases). So, the area is the limit of the above sum as n approaches infinity: The middle expression is called the definite integral of f from a to b; f is called the integrand; a and b are limits of integration; dx shows that x is the variable of integration.