Question

use geometric formulas to find the unsigned area between the graph of ( y = f(x) ) and the x - axis over the indicated interval.

( f(x)=x + 9;-2,10 )

the area between ( f(x) ) and the x - axis is ( square ).
(type an exact answer, using ( pi ) as needed.)

Explanation:

Step1: Find the function values at endpoints

When \(x = - 2\), \(y=f(-2)=-2 + 9=7\).
When \(x = 10\), \(y=f(10)=10 + 9=19\).

Step2: Use the trapezoid area formula

The formula for the area of a trapezoid is \(A=\frac{(b_1 + b_2)h}{2}\), where \(b_1\) and \(b_2\) are the lengths of the parallel sides and \(h\) is the height.
Here, \(b_1 = 7\), \(b_2=19\), and \(h=10-(-2)=12\).
Substitute into the formula: \(A=\frac{(7 + 19)\times12}{2}\).
First, calculate \(7+19 = 26\). Then \(26\times12=312\). And \(\frac{312}{2}=156\).

Answer:

\(156\)