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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Chapter
Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Use a graph to give a rough estimate of the area of the region that lies under the curve y = x x , 0 x 4 . Then find the exact area.

To determine

The value of the area under the curve.

Explanation

Given information:

The equation of curve is y=xx (1)

The region lies between x=0 and x=4.

Calculation:

To find the value of y by using Equation (1) as shown below:

Substitute 0 for x in Equation (1).

y=00=0

Substitute 1 for x in Equation (1).

y=11=1

Similarly calculate the values of y for different x values as shown in Table (1).

xy
00
11.00
22.83
35.20
48.00

Plot the curve using the values from Table (1).

Draw the diagram for the equation of curve as shown in Figure (1).

Refer Figure (1)

Here, area of the shaded portion 1 is A1, area of the shaded portion 2 is A2, area of the shaded portion 3 is A3 and the area of the shaded portion 4 is A4,

To find the area under the graph using Figure (1) as shown below:

Area=A1+A2+A3+A4=(12×1×1)+(12×1×(1+2.83))+(12×1×(2

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