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

The integral, therefore, is:

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

The answer is -3.