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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

If

f ( x ) = cos x 0 x 3 π / 4

evaluate the Riemann sum with n = 6, taking the sample points to be left endpoints. (Give your answer correct to six decimal places.) What does the Riemann sum represent? Illustrate with a diagram.

To determine

To evaluate: The Riemann sum for the function f(x)=cosx for six sub intervals using left end-points.

To represent: The Riemann sum for the function f(x)=cosx.

Explanation

Given information:

The function f(x)=cosx lies between 0 and 3π4. Hence, the limits are a=0 and b=3π4.

Number of rectangles is n=6.

Calculation:

Find the width (Δx) using the relation:

Δx=ban (1)

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

Substitute 3π4 for b, 0 for a, and 6 for n in Equation (1).

Δx=(3π406)=π8

Determine left end points xi1 using the relation:

xi1=a+(i1)Δx (2)

Calculate the value of xi1 for six subintervals using Equation (2).

Substitute 0 for a, 1 for i, and π8 for Δx in Equation (2).

x0=0+(0×π8)=0

Substitute 0 for a, 2 for i, and π8 for Δx in Equation (2).

x1=0+(1×π8)=π8

Substitute 0 for a, 3 for i and π8 for Δx in Equation (2).

x2=0+2×π8=2π8

Substitute 0 for a, 4 for i, and π8 for Δx in Equation (2).

x3=0+3×π8=3π8

Substitute 0 for a, 5 for i, and π8 for Δx in Equation (2).

x4=0+4×π8=4π8

Determine the value of f(xi1) using the relation:

f(xi1)=cos(xi1) (3)

Calculate the value of f(xi1) for six subintervals using Equation (3).

Substitute 1 for i and 0 for x0 in the Equation (3).

f(x0)=cos(x0)=cos(0)=1

Substitute 2 for i and π8 for x1 in the Equation (3).

f(x1)=cos(x1)=cos(π8)=0.923880

Substitute 3 for i and 2π8 for x2 in the Equation (3).

f(x2)=cos(x2)=cos(2π8)=0.707107

Substitute 4 for i and 3π8 for x3 in the Equation (3)

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