   Chapter 4.2, Problem 31E

Chapter
Section
Textbook Problem

# Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit. ∫ 0 π sin 5 x   d x

To determine

To express:

The integral 0πsin5x dx as a limit of the sums then evaluate by using a computer’s algebra system to find both the sum and the limit.

Explanation

1) Concept:

Use theorem (4) to express the given integral as a limit of the sums then use a computer’s algebra system to find both the sum and the limit.

Theorem (4):

If f is integrable on [a, b] then

abfxdx=limni=1nfxi x

Where x= b - an and xi=a+i x

Also, i=1nf(xi*)x is the Riemann sum formula.

2) Given:

0πsin5x dx

3) Calculation:

Comparing the given integral with theorem (4), set

a=0, b=π and fx=sin5x

Substituting the values of a and b in x,

x= b-an

x= πn

Now, find xi.

xi=a+i x

xi=πin

By using theorem (4) and substituting the values of x and xi, fxi=sin5xi=sin5

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