question the function f(x) = 5/3 x - 5 is gra...
question the function f(x) = 5/3 x - 5 is graphed below. use geometric formulas to evaluate the following definite integral. ∫₄⁸ (5/3 x - 5) dx enter an exact answer.
Answer
# Explanation:
## Step1: Find function values at endpoints
When \(x = 4\), \(y_1=\frac{5}{3}\times4 - 5=\frac{20}{3}-5=\frac{20 - 15}{3}=\frac{5}{3}\). When \(x = 8\), \(y_2=\frac{5}{3}\times8 - 5=\frac{40}{3}-5=\frac{40 - 15}{3}=\frac{25}{3}\).
## Step2: Recognize geometric - shape
The definite integral \(\int_{4}^{8}(\frac{5}{3}x - 5)dx\) represents the area between the line \(y=\frac{5}{3}x - 5\), \(x = 4\), \(x = 8\) and the \(x\) - axis. The region 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, \(b_1=\frac{5}{3}\), \(b_2=\frac{25}{3}\), and \(h=8 - 4 = 4\). Then \(A=\frac{1}{2}(\frac{5}{3}+\frac{25}{3})\times4\).
## Step4: Simplify the expression
First, add the fractions inside the parentheses: \(\frac{5}{3}+\frac{25}{3}=\frac{5 + 25}{3}=\frac{30}{3}=10\). Then, \(A=\frac{1}{2}\times10\times4=20\).
# Answer:
20