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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
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 beneath the given curve. Then find the exact area.

y = x 3 , 0 x 27

To determine

The area under the curve y=x3.

Explanation

Given information:

The equation of curve is y=x3 (1)

The region lies between x=0 to x=27.

Calculation:

Find y value using the curve function:

Substitute 0 for x in Equation (1).

y=03=0

Hence, the co-ordinates (x,y) is (0,0).

Substitute 1 for x in Equation (1).

y=x3=13y=1

Hence, the co-ordinates (x,y) is (1,1).

Similarly calculate for remaining values and are shown in Table (1).

Table (1) shows the co-ordinates of (x,y) for the interval x=0 to x=27.

xy
00.00
11.00
21.26
31.44
41.59
51.71
61.82
71.91
82.00
92.08
102.15
112.22
122.29
132.35
142.41
152.47
162.52
172.57
182.62
192.67
202.71
212.76
222.80
232.84
242.88
252.92
262.96
273.00

Draw the diagram for the curve function y=x3 using the calculated values in Table (1).

Show the diagram for the curve function y=x3 as in Figure (1).

Refer to Figure (1).

The rough estimate of area can be obtianed by splitting the region bounded by the curve

The area under the curve is calculated approximately by deducting the area of A2 and A3 from A1

Find the rectangular area (A1) as shown below:

A1=bh=27×3=81

Find the traingular area (A2) as shown below:

A2=12×b×h=12×27×1

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