Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

Evaluate the integral by interpreting it in terms of areas

\(\int\limits_{\rm{0}}^{\rm{9}} {\left( {\frac{{\rm{1}}}{{\rm{3}}}{\rm{x - 2}}} \right){\rm{dx}}} {\rm{.}}\)

The value of integral is \({\rm{ - 4}}{\rm{.5}}\)

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Objective of the question.

Consider the following integral:

\(\int\limits_{\rm{0}}^{\rm{9}} {\left( {\frac{{\rm{1}}}{{\rm{3}}}{\rm{x - 2}}} \right){\rm{dx}}} \)

The objective is to interpret the definite integral \(\int\limits_{\rm{0}}^{\rm{9}} {\left( {\frac{{\rm{1}}}{{\rm{3}}}{\rm{x - 2}}} \right){\rm{dx}}} \) is as the area of the region bounded between the function \({\rm{f(x) = }}\frac{{\rm{1}}}{{\rm{3}}}{\rm{x - 2}}\) and the axis over the interval \(\left( {{\rm{0,9}}} \right)\)

The region is as shown below:

Step 2: Given information.

The shaded region consists of two triangles.

Region A is a triangle with base length \({\rm{6}}\)and height \({\rm{2}}\)

So, the area of region A is as follow:

\({\rm{A = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{bh = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{(6)(2) = 6}}\)

Step 3: Area of Triangle.

Region B is a triangle with base length \({\rm{3}}\) and height \({\rm{1}}\)

So, the area of the region B is as follow:

\({\rm{B = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{bh = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{(3)(1) = 1}}{\rm{.5}}\)

Step 4: Find the total integral.

The total integral is the sum of the positive areas above the \({\rm{x}}\)axis and the negative area below the \({\rm{x}}\)axis.

Then,

\(\int\limits_{\rm{0}}^{\rm{9}} {\left( {\frac{{\rm{1}}}{{\rm{3}}}{\rm{x - 2}}} \right){\rm{dx = - A + B}}} {\rm{ = - 6 + 1}}{\rm{.5 = - 4}}{\rm{.5}}\)

Therefore,

The value of the integral is

\({\rm{ - 4}}{\rm{.5}}\)

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Evaluate the integral by interpreting it in terms of areas

\(\int\limits_{\rm{0}}^{\rm{9}} {\left( {\frac{{\rm{1}}}{{\rm{3}}}{\rm{x - 2}}} \right){\rm{dx}}} {\rm{.}}\)

Step by Step Solution

Step 1: Objective of the question.

Step 2: Given information.

Step 3: Area of Triangle.

Step 4: Find the total integral.

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Evaluate the integral by interpreting it in terms of areas

\(\int\limits_{\rm{0}}^{\rm{9}} {\left( {\frac{{\rm{1}}}{{\rm{3}}}{\rm{x - 2}}} \right){\rm{dx}}} {\rm{.}}\)

Step by Step Solution

Step 1: Objective of the question.

Step 2: Given information.

Step 3: Area of Triangle.

Step 4: Find the total integral.

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