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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

Find the area between the curves in Problems 19-22.

y = x 3  and  y = x  from  x = 1  to  x = 0

To determine

To calculate: The area under the graph of y=x3 and y=x from x=1 to x=0 using the definite integral method.

Explanation

Given Information:

The provided functions are y=x3 and y=x and range of x is 1x0.

Formula Used:

The area between two graphs:

If f(x)g(x) for all x in [a,b] (so that the graph of f doesn’t move below that of g), then the area of the region between the graphs of f and g and between x=a and x=b is given by:

A=ab[f(x)g(x)dx]

Calculation:

The provided functions are y=x3 and y=x and range of x is 1x0.

The point of intersection can be obtained by keeping them equal:

x3=xx3x=0x(x21)=0

Therefore, x=0 or x21=0.

That is, x=0 or x=1 or x=1

Substitute x=0 in the equation y=x.

y=x=0

Now, substitute x=1 in the equation y=x.

y=x=1

Now, substitute x=1 in the equation y=x.

y=x=1

Therefore, both graphs intersect at points (0,0), (1,1) and (1,1).

Consider the equation y=x3.

Substitute x=0 in y=x3.

y=(0)3=0

Substitute x=2 in y=x3.

y=(2)3=8

Substitute x=2 in y=x3.

y=(2)3=8

Summarized all points in a table,

x y (x, y)
0 0 (0,0)
2 8 (2,8)
2 8 (2,8)

Now, consider the equation y=x.

Substitute x=0 in y=x.

y=(0)=0

Substitute x=2 in y=x

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