   Chapter 13.1, Problem 7E ### 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

# In Problems 5-8, approximate the area under each curve by evaluating the function at the left-hand endpoints of the subintervals. f ( x ) = 9 − x 2  from  x = 1  to  x = 1 ; 4  subintervals

To determine

To calculate: The area under the graph of f(x)=9x2 from x=1 to x=3 using the 4 equal-intervals of the same size and use rectangles to find the exact area.

Explanation

Given Information:

The curve is f(x)=9x2 from x=1 to x=3 and 4 subintervals and the function is evaluated at left end points of the subintervals.

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 left-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:

Consider the curve f(x)=9x2 from x=1 to x=3 and 4 subintervals where the function is evaluated at left 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=1 to x=3. So, a=1 and b=3.

Thus,

Base of each rectangle=314=0.5

Thus, the 4 subintervals, each of length 0.5 are [1,1.5], [1.5,2], [2,2.5] and [2.5,3].

Since, there are 4 subintervals, the number of rectangles is 4.

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

Record these values in a table.

 Rectangle Base Left endpoint Height Area =base×height 1 0

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