<< Chapter < Page Chapter >> Page >
  1. Find an upper sum for f ( x ) = 10 x 2 on [ 1 , 2 ] ; let n = 4 .
  2. Sketch the approximation.
  1. Upper sum = 8.0313 .

  2. A graph of the function f(x) = 10 − x^2 from 0 to 2. It is set up for a right endpoint approximation over the area [1,2], which is labeled a=x0 to x4. It is an upper sum.

Got questions? Get instant answers now!

Finding lower and upper sums for f ( x ) = sin x

Find a lower sum for f ( x ) = sin x over the interval [ a , b ] = [ 0 , π 2 ] ; let n = 6 .

Let’s first look at the graph in [link] to get a better idea of the area of interest.

A graph of the function y = sin(x) from 0 to pi. It is set up for a left endpoint approximation from 0 to pi/2 and n=6. It is a lower sum.
The graph of y = sin x is divided into six regions: Δ x = π / 2 6 = π 12 .

The intervals are [ 0 , π 12 ] , [ π 12 , π 6 ] , [ π 6 , π 4 ] , [ π 4 , π 3 ] , [ π 3 , 5 π 12 ] , and [ 5 π 12 , π 2 ] . Note that f ( x ) = sin x is increasing on the interval [ 0 , π 2 ] , so a left-endpoint approximation gives us the lower sum. A left-endpoint approximation is the Riemann sum i = 0 5 sin x i ( π 12 ) . We have

A sin ( 0 ) ( π 12 ) + sin ( π 12 ) ( π 12 ) + sin ( π 6 ) ( π 12 ) + sin ( π 4 ) ( π 12 ) + sin ( π 3 ) ( π 12 ) + sin ( 5 π 12 ) ( π 12 ) = 0.863.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Using the function f ( x ) = sin x over the interval [ 0 , π 2 ] , find an upper sum; let n = 6 .

A 1.125

Got questions? Get instant answers now!

Key concepts

  • The use of sigma (summation) notation of the form i = 1 n a i is useful for expressing long sums of values in compact form.
  • For a continuous function defined over an interval [ a , b ] , the process of dividing the interval into n equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.
  • The width of each rectangle is Δ x = b a n .
  • Riemann sums are expressions of the form i = 1 n f ( x i * ) Δ x , and can be used to estimate the area under the curve y = f ( x ) . Left- and right-endpoint approximations are special kinds of Riemann sums where the values of { x i * } are chosen to be the left or right endpoints of the subintervals, respectively.
  • Riemann sums allow for much flexibility in choosing the set of points { x i * } at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.

Key equations

  • Properties of Sigma Notation
    i = 1 n c = n c
    i = 1 n c a i = c i = 1 n a i
    i = 1 n ( a i + b i ) = i = 1 n a i + i = 1 n b i
    i = 1 n ( a i b i ) = i = 1 n a i i = 1 n b i
    i = 1 n a i = i = 1 m a i + i = m + 1 n a i
  • Sums and Powers of Integers
    i = 1 n i = 1 + 2 + + n = n ( n + 1 ) 2
    i = 1 n i 2 = 1 2 + 2 2 + + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6
    i = 0 n i 3 = 1 3 + 2 3 + + n 3 = n 2 ( n + 1 ) 2 4
  • Left-Endpoint Approximation
    A L n = f ( x 0 ) Δ x + f ( x 1 ) Δ x + + f ( x n 1 ) Δ x = i = 1 n f ( x i 1 ) Δ x
  • Right-Endpoint Approximation
    A R n = f ( x 1 ) Δ x + f ( x 2 ) Δ x + + f ( x n ) Δ x = i = 1 n f ( x i ) Δ x

State whether the given sums are equal or unequal.

  1. i = 1 10 i and k = 1 10 k
  2. i = 1 10 i and i = 6 15 ( i 5 )
  3. i = 1 10 i ( i 1 ) and j = 0 9 ( j + 1 ) j
  4. i = 1 10 i ( i 1 ) and k = 1 10 ( k 2 k )

a. They are equal; both represent the sum of the first 10 whole numbers. b. They are equal; both represent the sum of the first 10 whole numbers. c. They are equal by substituting j = i 1 . d. They are equal; the first sum factors the terms of the second.

Got questions? Get instant answers now!

In the following exercises, use the rules for sums of powers of integers to compute the sums.

i = 5 10 i 2

385 30 = 355

Got questions? Get instant answers now!

Suppose that i = 1 100 a i = 15 and i = 1 100 b i = −12 . In the following exercises, compute the sums.

i = 1 100 ( a i + b i )

Got questions? Get instant answers now!

i = 1 100 ( a i b i )

15 ( −12 ) = 27

Got questions? Get instant answers now!

i = 1 100 ( 3 a i 4 b i )

Got questions? Get instant answers now!

i = 1 100 ( 5 a i + 4 b i )

5 ( 15 ) + 4 ( −12 ) = 27

Got questions? Get instant answers now!

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.

k = 1 20 100 ( k 2 5 k + 1 )

Got questions? Get instant answers now!

j = 1 50 ( j 2 2 j )

j = 1 50 j 2 2 j = 1 50 j = ( 50 ) ( 51 ) ( 101 ) 6 2 ( 50 ) ( 51 ) 2 = 40 , 375

Got questions? Get instant answers now!

j = 11 20 ( j 2 10 j )

Got questions? Get instant answers now!

k = 1 25 [ ( 2 k ) 2 100 k ]

4 k = 1 25 k 2 100 k = 1 25 k = 4 ( 25 ) ( 26 ) ( 51 ) 9 50 ( 25 ) ( 26 ) = −10 , 400

Got questions? Get instant answers now!
Practice Key Terms 8

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask