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 5.1, Problem 23E

(a)

To determine

To find: The relation between area, lower sum, and upper sum for an increasing function f.

Expert Solution

Answer to Problem 23E

The relation between area, lower sum, and upper sum for a function is Ln<A<Rn_ .

Explanation of Solution

Given information:

The curve f is an increasing continuous function.

Describe the relation between Ln , A, and Rn as below:

  • The increasing function f represents the lower sum (Ln) and an upper sum (Rn) . So Ln is an underestimate of area (A) and Rn is an overestimate of area (A).
  • The value of Rn is more than the area A and the value of Ln is less than the area A.

Therefore, the terms are related as Ln<A<Rn_ .

(b)

To determine

To prove: The relation RnLn=ban[f(b)f(a)] .

Expert Solution

Answer to Problem 23E

The relation RnLn=ban[f(b)f(a)] is proved.

Explanation of Solution

The expression for upper estimate of area is shown below:

Rn=f(x1)Δx+f(x2)Δx+...+f(xn)Δx (1)

Here, the right endpoint height of the first rectangle is f(x1) , the width of the interval is Δx , the right endpoint height of the second rectangle is f(x2) , and the right endpoint height of nth rectangle is f(xn) .

The expression for lower estimate of area is shown below:

Ln=f(x0)Δx+f(x1)Δx+...+f(xn1)Δx (2)

Here, the left endpoint height of the first rectangle is f(x0) , the left endpoint height of the second rectangle is f(x1) , and the left endpoint height of nth rectangle is f(xn1) .

Subtract Equation (2) from Equation (1) as shown below:

RnLn=[[f(x1)Δx+f(x2)Δx+...+f(xn)Δx][f(x0)Δx+f(x1)Δx+...+f(xn1)Δx]]=f(xn)Δxf(x0)Δx=Δx[f(xn)f(x0)] (3)

The width of the interval (Δx) is expressed as shown below:

Δx=ban (4)

Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.

Substitute b for xn , a for x0 , and ban for Δx in Equation (3).

RnLn=ban[f(b)f(a)]

Hence, the relation is proved.

Draw the diagram for an increasing function f with n rectangles as shown in Figure (1).

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 5.1, Problem 23E

Refer to Figure (1).

  • The difference in upper sum and lower sum (RnLn) represents the sum of areas of the shaded rectangles.
  • Slide the shaded rectangles to the left and stack it on the top of the leftmost shaded rectangle.
  • The stacked rectangles forms a rectangle with height f(b)f(a) and width Δx=ban .
  • Hence, the area is ban[f(b)f(a)] .

Therefore, the estimate RnLn is reassembled and forms a single rectangle with area of ban[f(b)f(a)] .

(c)

To determine

To prove: (RnA) is less than ban[f(b)f(a)] .

Expert Solution

Answer to Problem 23E

Yes, the expression is proved.

Explanation of Solution

Refer to part (a).

The value of Ln is less than the area A.

Therefore, it can be expressed as A>Ln . Also it is valid for the inequality RnA<RnLn .

Substitute ban[f(b)f(a)] for RnLn .

RnA<ban[f(b)f(a)]

Therefore, the expression is valid and it is proved.

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