the function f(x) = 5/3 x - 5 is graphed belo...
the function f(x) = 5/3 x - 5 is graphed below. use geometric formulas to evaluate the following definite integral. ∫₄⁶ (5/3 x - 5) dx enter an exact answer.
Answer
# Explanation:
## Step1: Find function values at endpoints
When \(x = 4\), \(f(4)=\frac{5}{3}\times4 - 5=\frac{20}{3}-5=\frac{20 - 15}{3}=\frac{5}{3}\). When \(x = 6\), \(f(6)=\frac{5}{3}\times6-5 = 10 - 5=5\).
## Step2: Recognize geometric shape
The definite - integral \(\int_{4}^{6}(\frac{5}{3}x - 5)dx\) represents the area between the line \(y=\frac{5}{3}x - 5\), the \(x\) - axis, \(x = 4\), and \(x = 6\). This area is a trapezoid.
## Step3: Apply trapezoid area formula
The area formula for a trapezoid is \(A=\frac{1}{2}(b_1 + b_2)h\), where \(b_1\) and \(b_2\) are the lengths of the parallel sides and \(h\) is the height. Here, \(b_1=\frac{5}{3}\), \(b_2 = 5\), and \(h=6 - 4 = 2\).
\[
\begin{align*}
A&=\frac{1}{2}(\frac{5}{3}+5)\times2\\
&=\frac{1}{2}(\frac{5 + 15}{3})\times2\\
&=\frac{1}{2}\times\frac{20}{3}\times2\\
&=\frac{20}{3}
\end{align*}
\]
# Answer:
\(\frac{20}{3}\)