BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 5, Problem 2RCC

(a)

To determine

To write: The definition for the definite integral of a continuous function from a to b.

Expert Solution

Explanation of Solution

Consider the function f which is continuous on interval [a,b].

Divide the interval into n subintervals of equal width, Δx=ban.

Let x0(=a),x1,x2,....,xn(=b) be the end and intermediate points of the subintervals.

Then, the definite interval is the sum of the area of the subintervals. It is expressed as follows:

abf(x)dx=limni=1nf(xi*)Δx

Here, xi* is any sample point in the ith subinterval [xi1,xi].

(b)

To determine

To find: The geometric interpretation of abf(x)dx if f(x)0.

Expert Solution

Explanation of Solution

The function f(x)0 represents the function in the first quadrant of the graph.

Sketch the curve f(x) in the first quadrantas shown in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5, Problem 2RCC , additional homework tip  1

From Figure 1, the function f is positive when f0.

The geometric interpretation of abf(x)dx refers that the total area which is the region under the graph y=f(x) and above the x-axis on the interval [a,b].

(c)

To determine

To find: The geometric interpretation of abf(x)dx, if f(x) have both positive and negative values.

Expert Solution

Explanation of Solution

The function f(x) consists both positive and negative values which means that the graph of y=f(x) forms the region above and below the x-axis.

The graph of function y=f(x) for both positive and negative value of f(x) is shown in the below Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5, Problem 2RCC , additional homework tip  2

From the definition from part (a), as f(x) takes both positive and negative values the limit of each Riemann sum is also positive, negative or zero.

Thus, the geometric interpretation of abf(x)dx refers the signed area under the graph y=f(x) above the x-axis and below the x-axis on the interval [a,b].

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