question the function f(x) = 5/3 x - 5 is graphed below. use geometric formulas to evaluate the following…

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.

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