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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 2  and  y = 4 x + 5

To determine

To calculate: The area under the graph of y=x2 and y=4x+5 using the definite integral method.

Explanation

Given Information:

The provided functions are y=x2 and y=4x+5.

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=x2 and y=4x+5.

The points can be obtained by keeping them equal:

x2=4x+5x24x5=0x25x+x5=0x(x5)+1(x5)=0

Further simplify,

x(x5)+1(x5)=0(x+1)(x5)=0

Therefore, x=1 or x=5.

Substitute x=1 in the equation y=x2.

y=x2=(1)2=1

Now, substitute x=5 in the equation y=x2.

y=x2=(5)2=25

Therefore, both graphs intersect at points (1,1) and (5,25).

Consider the equation y=x2.

The graph is obtained as follows:

Substitute x=0 in y=x2.

y=(0)2=0

Substitute x=1 in y=x2.

y=(1)2=1

Substitute x=3 in y=x2.

y=(3)2=9

Summarized all points in a table,

x y (x,y)
0 0 (0,0)
1 1 (1,1)
3 9 (3,9)

Now, consider the equation y=4x+5,

Substitute x=0 in y=4x+5.

y=4(0)+5=5

Substitute x=1 in y=4x+5.

y=4(1)+5=9

Substitute x=3 in y=4x+5

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Sect-13.1 P-9ESect-13.1 P-10ESect-13.1 P-11ESect-13.1 P-12ESect-13.1 P-13ESect-13.1 P-14ESect-13.1 P-15ESect-13.1 P-16ESect-13.1 P-17ESect-13.1 P-18ESect-13.1 P-19ESect-13.1 P-20ESect-13.1 P-21ESect-13.1 P-22ESect-13.1 P-23ESect-13.1 P-24ESect-13.1 P-25ESect-13.1 P-26ESect-13.1 P-27ESect-13.1 P-28ESect-13.1 P-29ESect-13.1 P-30ESect-13.1 P-31ESect-13.1 P-32ESect-13.1 P-33ESect-13.1 P-34ESect-13.1 P-35ESect-13.1 P-36ESect-13.1 P-37ESect-13.1 P-38ESect-13.1 P-39ESect-13.1 P-40ESect-13.2 P-1CPSect-13.2 P-2CPSect-13.2 P-1ESect-13.2 P-2ESect-13.2 P-3ESect-13.2 P-4ESect-13.2 P-5ESect-13.2 P-6ESect-13.2 P-9ESect-13.2 P-10ESect-13.2 P-11ESect-13.2 P-12ESect-13.2 P-7ESect-13.2 P-8ESect-13.2 P-23ESect-13.2 P-24ESect-13.2 P-13ESect-13.2 P-14ESect-13.2 P-15ESect-13.2 P-16ESect-13.2 P-17ESect-13.2 P-18ESect-13.2 P-19ESect-13.2 P-20ESect-13.2 P-21ESect-13.2 P-22ESect-13.2 P-25ESect-13.2 P-26ESect-13.2 P-27ESect-13.2 P-28ESect-13.2 P-29ESect-13.2 P-30ESect-13.2 P-31ESect-13.2 P-32ESect-13.2 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