   Chapter 5, Problem 108RE ### 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 105-110, use the Midpoint Rule with n =   4 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 ) = x 3 + 8                      [ − 2 , 2 ]

To determine

To calculate: The area of the region bounded by the graph of the function f(x)=x3+8 where the intervals are [2,2] use midpoint rule with n as 4 and sketch the region.

Explanation

Given Information:

The provided function is f(x)=x3+8 and the intervals is intervals are [2,2].

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)=x3+8

The intervals are [2,2] with n=4.

Now divide the provided interval into 4 subparts as shown below:

Δx=2+24=1

Therefore the 4 subintervals are,

[2,1],[1,0],[0,1],[1,2]

Now find the mid points of these intervals because each subinterval has a width of 1.

Therefore, the mid points of these interval are shown below,

32,12,12 and 32

Here, the mid points 32,12,12 and 32 lies in the middle of [2,1],[1,0],[0,1],[1,2] intervals respectively

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