Answered: y. AG) = f(21 + 3) đt f(t) = 21 + 3…

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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Question

Verify that the area function y = A(x), for x >= 2. gives the
correct area when x = 6 and x = 10

y.
AG) = f(21 + 3) đt
f(t) = 21 + 3
A(x)
х

Expert Solution

Step 1

We have to verify that area function $y = A (x)$ for $x \geq 2$ gives the correct area when $x = 6 a n d x = 10$

Step 2

When $x = 6$ then area, given by area function between $x = 2 a n d x = 6$ is

$\begin{array}{rcl} A (6) & = & \int_{2}^{6} (2 t + 3) d t \\ = & {[\frac{2 t^{2}}{2} + 3 t]}_{2}^{6} \\ = & {[t^{2} + 3 t]}_{2}^{6} \\ = & [(36 + 18) - (4 + 6)] \\ = & 44 \end{array}$

We know the formula of area for trapezoid:

$A = \frac{1}{2} (a + b) h$

Where, $a a n d b$ represent the length of parallel sides of trapezoid and $h$ denotes the distance between parallel sides.

When $x = 2$ then value of $y$ is

$\begin{array}{rcl} y & = & 2 \times 2 + 3 \\ = & 7 \end{array}$

When $x = 6$ then value of $y$ is

$\begin{array}{rcl} y & = & 2 \times 6 + 3 \\ = & 15 \end{array}$

When $x = 6$ the area between $x = 2 a n d x = 6$ is trapezoid which have parallel sides length $7 a n d 15$ and distance between parallel sides is $4$ .

So, area between $x = 2 a n d x = 6$ is:

$\begin{array}{rcl} A & = & \frac{1}{2} (7 + 15) \times 4 \\ = & \frac{22 \times 4}{2} \\ = & 44 \end{array}$

So, area function is verified for $x = 6$

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