   Chapter 5, Problem 110RE ### 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 ) = 3 x x 3 + 2      [0,4]

To determine

To calculate: The area of the region bounded by the graph of the function f(x)=3xx3+2 where the intervals are [0,4] use midpoint rule with n as 4 and sketch the region.

Explanation

Given Information:

The provided function is f(x)=3xx3+2 and the intervals is intervals are [0,4]

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)=3xx3+2

The intervals are [0,4] with n=4.

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

Δx=404=1

Therefore the 4 subintervals are,

[0,1],[1,2],[2,3],[3,4]

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:

12,32,52 and 72

Here the mid points 12,32,52 and 72 lies in the middle of [0,1],[1,2],[2,3],[3,4] intervals respectively

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