   Chapter 13, Problem 4RE ### Mathematical Applications for the ...

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

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

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

# Use rectangles to find the exact area under the graph of y  –  3 x 2 from x = 0 to x = 1. Use n equal subintervals.

To determine

To calculate: The area under the graph of y=3x2 from x=0 to x=1 using the n equal-intervals of the same size and use rectangles to find the exact area.

Explanation

Given Information:

The provided function is y=3x2 from x=0 to x=1.

Formula Used:

The base of the rectangles to approximate the area is ban where the interval [a,b] on which function is defined is divided into n subintervals.

The height of the rectangles is the value of the function calculated at the right-hand end point of the interval containing the base.

The area of a rectangle is base×height.

The approximated area under the curve is the sum of the areas of each rectangle.

The value of the sum k=1nk3=(n(n+1)2)2.

Calculation:

The curve is y=3x2 from x=0 to x=1 and the interval [0,1] is divided into n subintervals and the function is evaluated at right-hand end points of the subintervals.

The base of the rectangles to approximate the area is ban where the interval [a,b] on which function is defined is divided into n subintervals.

Since, the function is defined from x=0 to x=1. So, a=0 and b=1.

Thus,

Base of each rectangle=10n=1n

Thus, the n subintervals, each of length 1n, are [0,1n], [1n,2n],,[n1n,1].

Since, there are n subintervals, the number of rectangles is n.

Recall that the height of the rectangles is the value of the function calculated at the right-hand end point of the interval containing the base.

Since, the right-hand end point of the first subinterval [0,1n] is 1n.

Thus, the height of the first rectangle is,

y=3x2=3(1n)2=3n2

Recall that the area of a rectangle is base×height.

Thus, the area of the first rectangle is,

Area=base×height=1n×3n2=3n3

Do a similar calculation to find the area of all n rectangles and record these values in a table.

 Rectangle Base Right endpoint Height Area=base×height 1 1n x1=1n y=3(1n)2=3n2 1n×3n2=3n3 2 1n x2=2n y=3(2n)2=12n2 1n×12n2=12n3 3 1n x3=3n y=3(3n)2=27n2 1n×27n2=27n3 ⋮ ⋮ ⋮ ⋮ ⋮ n 1n xn=nn=1 y=3(1)2=3 1n×3=3n

Recall that the approximated area under the curve is sum of the areas of each rectangle

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