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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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BuyFindarrow_forward

Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem

Comparing Riemann Sums Consider a triangle of area 2 square units bounded by the graphs of y = x, y = 0, and x = 2.

(a) Sketch the graph of the region.

(b) Divide the interval [0, 2] into n equal subintervals and show that the endpoints of the subintervals are

0 < 1 ( 2 n ) < . . . < ( n 1 ) ( 2 n ) < n ( 2 n ) .

(c) Show that the left Riemann sum is

S L = i = 1 n [ ( i 1 ) ( 2 n ) ] ( 2 n ) .

(d) Show that the right Riemann sum is

S R = i = 1 n [ i ( 2 n ) ] ( 2 n ) .

(e) Use a graphing utility and the program in Example 4 to complete the table.

n 5 10 50 100
Left sum, SL
Left sum, SR

(f) Show that lim n S L = lim n S R = 2.

(a)

To determine

To graph: The region bounded by the graphs of y=x, y=0 and x=2.

Explanation

Given information:

The area of the triangle is 2 square units and the equations are y=x, y=0 and x=2.

Graph:

For equation y=x, it is a straight line.

Put x=0 in the equation y=x.

y=0

Put x=2 in the equation y=x.

y=2

The table show the plotted points of line y=x

(b)

To determine

The interval [0,2] into n equal subintervals and show that the endpoints of the subintervals are 0<1(2n)<<(n1)(2n)<n(2n)

(c)

To determine

To prove: The left Riemann sum is SL=i=1n[(i1)(2n)(2n)

(d)

To determine

To prove: The right Riemann sum is SR=i=1n[i(2n)(2n)

(e)

To determine

To fill: The table shown below by a graphing utility.

(f)

To determine

To prove: The limnSL=limnSR=2

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