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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

(a) Estimate the area under the graph of f(x) = 1 + x2 from x = −1 to x = 2 using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles.

(b) Repeat part (a) using left endpoints.

(c) Repeat part (a) using midpoints.

(d) From your sketches in parts (a)–(c), which appears to be the best estimate?

(a)

To determine

To find:

The area under the graph using right endpoints and three rectangles and then area under the graph using right endpoints and six rectangles.

Explanation

Given:

The curve function is f(x)=1+x2.

The region lies between x=1 and x=2. So the limits are a=1 and b=2.

Calculation:

Draw the graph for the function f(x)=1+x2 with three rectangles using the right endpoints as shown in Figure (1).

The expression to find the estimate of the areas of n rectangles (Rn) using right endpoints 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.

Find the area estimate for three rectangles with right end points.

Substitute 2 for b, -1 for a, and 3 for n in Equation (2).

Δx=2+13=1

From Figure (1), take the right endpoint height of the first rectangle’s f(x1) value as 1, the right endpoint height of the second rectangle’s f(x2) value as 2, and the right endpoint height of the third rectangle’s f(x3) value as 5.

Substitute 3 for n, 1 for f(x1), 1 for Δx, 2 for f(x2), and 5 for f(x3) in Equation (1)

(b)

To determine

The area under the graph using left endpoints and three rectangles and then area under the graph using left endpoints and six rectangles.

(c)

To determine

The area under the graph using midpoints and three rectangles and then area under the graph using midpoints and six rectangles.

(d)

To determine

To find:

The best estimate.

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