   Chapter 4.2, Problem 23E

Chapter
Section
Textbook Problem

# Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. ∫ − 2 0 ( x 2 + x )   d x

To determine

To evaluate:

The integral -20x2+x dx by using the form of the definition of the integral given in the theorem (4).

Explanation

1) Concept:

Use the form of the definition of the integral given in the theorem (4).

Theorem (4):

If f is integrable on [a, b] then

abfxdx=limni=1nfxi x

where x= b - an and xi=a+i x

2) Formula:

i)i=1ni2= nn+1(2n+1)6

ii)i=1ni= n(n+1)2

iii)i=1ncai=ci=1naiwhere c is a constant

iv)i=1n(ai-bi)=i=1nai-i=1nbi

v)i=1nc =nc

3) Given:

-20x2+x dx

4) Calculation:

Compare the given integral with the theorem (4) that gives

a=-2, b=0 and fx=x2+x

Substituting value of a and b in x,

x= b-an

x= 0-(-2)n

x= 2n

Now, find xi.

xi=a+i x

xi=-2+2in

By using the theorem (4),

-20x2+x dx= limni=1nfxi x

Substitute the values of x and xi into the formula.

-20x2+x dx=limni=1nf-2+2in2n

By using the formula (iii),

-20x2+x dx= limn2ni=1nf-2+2in

= limn2ni=1n-2+2i

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