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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Surface Area Use the Midpoint Rule to estimate the surface area of the swamp shown in the figure.

Chapter 5, Problem 115RE, Surface Area Use the Midpoint Rule to estimate the surface area of the swamp shown in the figure.

To determine

To calculate: The surface area of the swamp shown below in the figure:

Calculus: An Applied Approach (MindTap Course List), Chapter 5, Problem 115RE

Explanation

Given information:

The provided figure is shown below:

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)]

Fundamental theorem of calculus:

abf(x)dx=F(x)]ab=F(b)F(a)

Calculation:

Consider the figure.

The lake is divided into 9 parts and the width of each part is 20 ft therefore the total width is:

55+70+60+60+72+64+57+54=180

Thus, Δx=180 and n=9

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