   Chapter 5, Problem 113RE ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Using the Midpoint Rule In Exercises 111-114, use the Midpoint Rule with n = 6 to approximate the area of the region bounded by the graph of f and the x-axis over the interval. Sketch the region. Function Interval f ( x ) = 3 x 2 + 1               [ − 6 , 6 ]

To determine

To calculate: The area of the region bounded by the graph of the function f(x)=3x2+1 where the intervals are [6,6] use midpoint rule with n as 6 and sketch the region.

Explanation

Given Information:

The provided function is f(x)=3x2+1 and the intervals is intervals are [6,6]

Formula used:

Steps to solve a definite integral abf(x)dx with the help of midpoint rule.

Step 1: For a given interval [a,b] divide it into n subintervals with having width of,

Δx=ban

Step 2: Evaluate the midpoint for the given subinterval. Midpoints={x1,x2,x3,xn}

Step 3: Find the value of f at each midpoint and make the sum as shown below:

abf(x)dxban[f(x1)+f(x2)+f(x3)++f(xn)]

Calculation:

Consider the function,

f(x)=3x2+1

The intervals are [6,6] with n=6

Now divide the provided interval into 6 subparts as shown below,

Δx=6+66=2

Therefore the 6 sub intervals are,

[6,4],[4,2],[2,0],[0,2],[2,4],[4,6]

Now find the mid points of these intervals because each subinterval has a width of 2. Therefore, the mid points of these interval are shown below,

5,3,1,1,3 and 5

Here the mid points 5,3,1,1,3 and 5 lies in the middle of [6,4],[4,2],[2,0],[0,2],[2,4],[4,6] intervals respectively

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