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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 5, Problem 105RE
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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BuyFindarrow_forward

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     [0,2] 

To determine

To calculate: The area of the region bounded by the graph of the function f(x)=x3 where the intervals are [0,2] by the use of midpoint rule with n as 4 and also sketch the region.

Explanation

Given Information:

The provided function is f(x)=x3 and the intervals is intervals are [0,2]

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=b−an

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)dx≈b−an[f(x1)+f(x2)+f(x3)+⋯+f(xn)]

Calculation:

Consider the provided function,

f(x)=x3

The intervals are [0,2] with n=4

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

Δx=2−04=12

Therefore the 4 subintervals are,

[0,12],[12,1],[1,32],[32,2]

Now find the mid points of these intervals because each subinterval has a width of 12. Therefore, the mid points of these interval are shown below,

14,34,54 and 74

Here the mid points 14,34,54 and 74 lies in the middle of [0,12],[12,1],[1,32],[32,2] intervals respectively

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True or False: is conservative.

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Briefly explain how a participant characteristic, such as personality, could be a confounding variable in a bet...

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Compute the price relatives for the four stocks making up the Boran index in exercise 20. Use the weighted aggr...

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Multiply (4x2+3)(4x23).

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Describe the difference between qualitative and quantitative research and recognize examples of each.

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The paper Mood Food: Chocolate and Depressive Symptoms in a Cross-Sectional Analysis (Archives of Internal Medi...

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Workers in several industries were surveyed to determine the proportion of workers who feel their industry is u...

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