Question

The graph of f consists of line segments, as shown in the figure. Evaluate each definite integral by using geometric formulas.

Answer 1

To find the definite integral between 5 and 11, we will add the areas of triangles A, B, and C. The area of Triangle B will be negative because it is below the x-axis. Remember that the area of a triangle equals base x height divided by 2.

$\begin{gathered} Area\text{ }of\text{ }A=\frac{1(1)}{2} \\ \\ Area\text{ }of\text{ }A=\frac{1}{2} \end{gathered}$$\begin{gathered} Area\text{ }of\text{ }B=\frac{4(2)}{2} \\ \\ Area\text{ }of\text{ }B=4 \end{gathered}$$\begin{gathered} Area\text{ }of\text{ }C=\frac{1(1)}{2} \\ \\ Area\text{o}f\text{C}=\frac{1}{2} \end{gathered}$

The integral, therefore, is:

$\begin{gathered} Total=\frac{1}{2}+(-4)+\frac{1}{2} \\ \\ Total=-3 \end{gathered}$

The answer is -3.