   Chapter 5.1, Problem 4E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# (a) Estimate the area under the graph of f(x) = sin x from x = 0 to x = π/2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?(b) Repeat part (a) using left endpoints.

(a)

To determine

To Analyze:

Whether the estimate is an underestimate or an overestimate using right endpoints.

Explanation

Given information:

The curve function is f(x)=sinx.

The region lies between x=0 and x=π2. So, the limits are a=0 and b=π2.

The number of rectangles n=4.

Draw the graph for the function f(x)=sinx as shown below in Figure (1):

Draw four approximating rectangles using right end points for the function f(x)=sinx as shown below in Figure (2):

The expression to find the lower estimate of areas of n rectangles (Rn) 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 is Δx, the right endpoint height of the second rectangle is f(x2), and the right endpoint height of nth rectangle is f(xn).

Find the width (Δx) using the relation:

Δx=ban (2)

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

Substitute π2 for b, 0 for a, and 4 for n in Equation (2)

(b)

To determine

To Analyze:

Whether the estimate is an underestimate or an overestimate using right endpoints.

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