We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.
Definite Integrals

Definite Integrals

Right-Hand Sum Exercises

Example 1

Let R be the region between the graph of f(x) = x2+ 1 and the x-axis on the interval [0, 4].

  • Draw R and the 8 rectangles that result from using a right-hand sum with 8 sub-intervals to approximate the area of R.
  • Use a Right-Hand Sum with 8 sub-intervals to approximate the area of R.
  • Is your approximation an under-estimate or an over-estimate to the actual area of R?

Example 2

Let S be the area between the graph of y = f (x) = 2x and the x-axis on the interval [1, 6].

  • Draw S. 
  • Use a Right-Hand Sum with 2 subintervals to approximate the area of S. Draw S and the rectangles used in this Right-Hand Sum on the same graph.
  • Use a Right-Hand Sum with 5 subintervals to approximate the area of S. Draw S and the rectangles used in this Right-Hand Sum on the same graph. 
  • Are your approximations in parts bigger or smaller than the actual area of S?

Example 3

Let W be the area between the graph of  and the x-axis on the interval [1, 4]. 

  • Draw W.
  • Use a Right-Hand Sum with 3 subintervals to approximate the area of W. Draw W and the rectangles used in this Right-Hand Sum on the same graph.
  • Use a Right-Hand Sum with 6 subintervals to approximate the area of W. Draw W and the rectangles used in this Right-Hand Sum on the same graph. 
  • Are your approximations bigger or smaller than the actual area of W?

Example 4

The table below shows some values of the increasing function f(x).

  • Use a right-hand sum with one sub-interval to estimate the area between the graph of f and the x-axis on the interval [2, 8].
  • Use a right-hand sum with three sub-intervals to estimate the area between the graph of f and the x-axis on the interval [2, 8].
  • Are your answers over- or under-estimates of the actual area between the graph of f and the x-axis on the interval [2, 8]?

Example 5

Some values of the decreasing function g are given in the table below:

  • Use a right-hand sum with 3 sub-intervals to estimate the area between the graph of g and the x-axis on the interval [-1, 2].
  • Use a right-hand sum with 2 sub-intervals to estimate the area between the graph of g and the x-axis on the interval [-1, 2].
  • Are your answers over- or under-estimates for the actual area between the graph of g and the x-axis on the interval [-1, 2]?

Example 6

  • Let W be the region between the graph of f and the x-axis on the interval [-20, 20].

Use a right-hand sum with 4 sub-intervals to estimate the area of W.

Example 7

  • Let Z be the region between the graph of g and the x-axis on the interval [-4, 0].

  • Use a right-hand sum with 2 sub-intervals to estimate the area of Z.
  • Use a right-hand sum with 4 sub-intervals to estimate the area of Z.
  • Are your answers over- or under- estimates for the area of Z?

Example 8

Let f(x) = x2 + 6x + 9. Use a right-hand sum with 6 sub-intervals to estimate the area between the graph of f and the x-axis on the interval [-6,-3].

Example 9

Let f (x) = -x2 + 2x + 8. Use a right-hand sum with 8 sub-intervals to estimate the area between the graph of f and the x-axis on the interval [0, 4].

Example 10

Let g be a function with values given by the table below. Use a right-hand sum with 3 sub-intervals to estimate the area between the graph of g and the x-axis on the interval [0, 12]. 

Example 11

Let h be a function with values given by the table below. Use a right-hand sum with 9 sub-intervals to estimate the area between the graph of h and the x-axis on the interval [-9, 9]. 

Example 12

The function f (x) on the interval [0, 30] is graphed below. Use a right-hand sum with 3 sub-intervals to estimate the area between the graph of f and the x-axis on this interval.

Example 13

Use a right-hand sum with the sub-intervals indicated by the data in the table to estimate the area between the graph of f and the x-axis on the interval [-10,1].

Advertisement