9) Given the function f(x)=-3x+2 , find the net signedx-axis over the interval [-5,3].area between f(x) and the

Answered: 9) Given the function f(x)=-3x+2 , find…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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9) Given the function f(x)=-3x+2 , find the net signed
x-axis over the interval [-5,3].
area between f(x) and the

Expert Solution

Step 1

Given function is

$f (x) = - 3 x + 2$

We have to find the area of the region bounded by the given function and the x-axis in the interval [-5, 3]

Step 2

$f (x) = - 3 x + 2 f (- 5) = - 3 \times (- 5) + 2 = 15 + 2 = 17 f (3) = - 3 \times (3) + 2 = - 9 + 2 = - 7$

The net signed area = Positive signed area - (-negative signed area)

Positive signed area

=Area above the x-axis

= Area of $△ A B C$

$= \frac{1}{2} \times b a s e \times h e i g h t = \frac{1}{2} \times (5 + \frac{2}{3}) \times 17 = \frac{1}{2} \times \frac{17}{3} \times 17 = \frac{{(17)}^{2}}{6} = \frac{289}{6}$

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