In Section 14.3, you will learn how to find the area between two curves. Suppose you have two functions f(x) and g(x). Also assume that the higher curve is f(x). We are interested in finding the area from a point x = a to x = b between the two curves. We can do this by finding the area below f(x) and above the x-axis,

$latex displaystyle intlimits_{a}^{b}{f(x),dx}$

and subtracting the area below g(x) and above the x-axis,

$latex displaystyle intlimits_{a}^{b}{g(x),dx}$

Alternately, we can subtract the functions first and then find the area,

This process for reversing the Product Rule for Derivatives is called Integration by Parts . It is covered in Section 14.2. In Integration by Parts, the integrand (the thing you are finding the antiderivative of) is written as a product. One piece is thought of as u and the other part v‘. The formula then says

$latex \int{u{v}’ dx=uv-\int{v{u}’ dx}}$

Below are several examples that students worked out.

Choose the expression for u. This is generally the inside part of a composition in the integrand. Use the derivative to find an other expression for du.

Match the integrand with u and du. All variables in the original integrand must change to u.

Change the integrand so that it is written in terms of u.

Work out the antiderivative in terms of u.

Put in the expression for u so that the antiderivative is written in terms of the original variable.

Now let’s look at the examples carried out in class.