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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) Find an approximation to the integral 0 4 ( x 2 3 x ) d x using a Riemann sum with right endpoints and n = 8.

(b) Draw a diagram like Figure 3 to illustrate the approximation in part (a).

(c) Use Theorem 4 to evaluate 0 4 ( x 2 3 x ) d x .

(d) Interpret the integral in part (c) as a difference of areas and illustrate with a diagram like Figure 4.

a)

To determine

To find: an approximation to the integral 04(x23x)dx using a Riemann sum with right endpoints.

Explanation

Given:

The integral function is, 04(x23x)dx.

The number of sub intervals as n=8.

The upper limit is 4 and lower limit is 0.

Theorem 4: If f is integrable on [a, b], then

abf(x)dx=limni=1nf(xi)Δx

Where Δx=ban and xi=a+iΔx

Calculation:

The integral function is,

04(x23x)dx

The Riemann sum with right endpointsformula is,

abf(x)dx=i=1nf(xi*)Δx (1)

Calculate the subinterval (Δx) using the formula:

Δx=ban

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

Substitute 4 for b, 0 for a, and 8 for n.

Δx=408=0.5

Hence, the subinterval length is 0.5 for the limits [0,4].

The subinterval values are 0.5, 1.0, 1.5, 2.0, 2.5,3.0, 3.5,and 4.0for x1, x2, x3, x4, x5, x6, x7, and x8

Calculate the approximation to the integral using a Riemann sum with right endpoints:

Substitute 0.5 for Δx and 8 for n in Equation (1).

abf(x)dx=i=18[f(xi)]0.5=[f(x1)+f(x2)+f(x3)+f(x4)+f(x5)+f(x6)+f(x7)+f(x8)]0.5 (2)

Apply the subinterval values in Equation (3).

abf(x)dx=[f(0.5)+f(1.0)+f(1.5)+f(2.0)+f(2.5)+f(3.0)+f(3.5)+f(4.0)]0

b)

To determine

To draw: a diagram similar to figure 3 to illustrate the approximation in part (a).

c)

To determine

To evaluate: the integral 04(x23x)dx using Theorem 4.

d)

To determine

To draw: a diagram similar to figure 4 to illustrate the integral in part (c) as a difference of areas.

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