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 + 10 );
0,11
the area between ( f(x) ) and the x - axis is ( square ).
(type an exact answer, using ( pi ) as needed.)

Explanation:

Step1: Identify the shape

The function \(y = x + 10\) is a linear function. Over the interval \([0,11]\), the graph of \(y=x + 10\) and the \(x -\)axis forms a trapezoid. 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.

When \(x = 0\), \(y=f(0)=0 + 10=10\) (one parallel side \(b_1 = 10\)). When \(x = 11\), \(y=f(11)=11+10 = 21\) (another parallel side \(b_2=21\)). The height \(h\) of the trapezoid is the length of the interval, so \(h=11-0 = 11\).

Step2: Apply the trapezoid area formula

Substitute \(b_1 = 10\), \(b_2=21\), and \(h = 11\) into the formula \(A=\frac{(b_1 + b_2)h}{2}\).

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Answer:

\(\frac{341}{2}\)