   Chapter 7.7, Problem 1E

Chapter
Section
Textbook Problem

# Let I = ∫ 0 4 f ( x )   d x , where f is the function whose graph is shown.(a) Use the graph to find L 2 , R 2 , and M 2 .(b) Are these underestimates or overestimates of I?(c) Use the graph to find T 2 . How does it compare with I?(d) For any value of n, list the numbers L n , R n , M n , T n , and I in increasing order. To determine

(a)

To find:

L2,R2,M2 from the given graph.

Explanation

Given:

a=0b=4n=2Δx=x(ba2)

=42=2

Formulas used:

Ln=i=1nf(xi1)Δx

Rn=i=1nf(xi)ΔxMn=Δi=1nf(x¯i)

(where xi’s are endpoints of subintervals and xi- is midpoint)

Tn=Δx2[f(x0)+2i=1n1f(xi)+f(xn)]

There are two intervals, so width of the interval is 402=2. Thus the intervals are [0,2], [2,4]

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