   Chapter 4.2, Problem 26E

Chapter
Section
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.

To determine

Part (a):

To find:

An approximation to the integral 04(x2-3x) dx using the Riemann sum with the right endpoints and n=8

Explanation

Concept:

Use the Riemann sum formula for  the right endpoints.

Theorem (4):

If f is integrable on [a, b] then

abf(x)dx=limni=1nf(xi)x

where  x=b - an,n is the number of the subintervals and xi=a+ix

Given:

04(x2-3x) dx and n=8

Calculation:

A Riemann sum shall give an estimate of the given integral. The sum here has 8 terms and shall use right end points of each interval.

Here, a=0, b=4, n=8 and fx=x2-3x

So, the width of the subintervals x is

x= b-an

x= 4-08=48= 12

Use the width x=12 to form the intervals.

Thus, the intervals are 0,12, 12,1, 1,32,32,2,2,52 , 52,3, 3,72, 72,4

The right endpoints are x1=12, x2=1, x3=32, x4=2, x5=52 , x6=3,x7=72 and x8=4

By using the Riemann sum formula,

R8=i

To determine

Part (b):

To draw:

A diagram like the figure 3 to illustrate the approximation in part (a)

To determine

Part (c):

To evaluate:

The integral 04x2-3x dx byusing theorem (4).

To determine

Part (d):

To interpret:

The integral in part (c) as a difference of the areas and illustrate with a diagram like the figure (4).

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