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Introduction to Definite Integrals - At A Glance:

There is much debate about who is more awesome...right-handers or south-paws. Would you want Shoeless Joe Jackson on your team, or Nomar Garciaparra? That's why it's better to have a switch hitter like Chipper Jones at bat; we don't have to pick.

In calculus, right-hand sums are similar to left-hand sums. However, instead of using the value of the function at the left endpoint of a sub-interval to determine rectangle height, we use the value of the function at the right endpoint of the sub-interval.

As with left-hand sums we can find the values of the function that we need using formulas, tables, or graphs.

Right-Hand Sums with Formulas

If we're given a formula for the function, we can use this formula to calculate the value of the function at the right endpoint of each sub-interval.

Right-Hand Sums with Tables

In order to find a right-hand sum we need to know the value of the function at the right endpoint of each sub-interval. We can take a right-hand sum if we have a table that contains the appropriate function values.

Sample Problem

Some values of the decreasing function f ( x ) are given by the following table:

  • Use a Right-Hand Sum with 2 sub-intervals to estimate the area between the graph of f and the x-axis on the interval [0,4].
    We don't know what the function f looks like, but we know these points are part of it:

Dividing the interval [0,4] into 2 equally-sized sub-intervals gives us sub-intervals of length 2. The height of the rectangle on [0,2] is f ( 2 ) = 17, so the area of this rectangle is

height ⋅ width = 17(2) = 34.

The height of the rectangle on [2,4] is f (4) = 3, so the area of this rectangle is

height ⋅ width = 3(2) = 6.

Adding the areas of these rectangles, we estimate the area between the graph of f and the x-axis on [0,4] to be

34 + 6 = 40.

  • Use a Right-Hand Sum with 4 sub-intervals to estimate the area between the graph of f and the x-axis on the interval [0,4].
    Answer. Dividing the interval [0,4] into 4 evenly-sized sub-intervals produces sub-intervals of length 1.

Sub-interval [0,1]: This rectangle has height f ( 0 ) = 20 and width 1, so its area is 20.

Sub-interval [1,2]: This rectangle has height f (1) = 18 and width 1, so its area is 18.

Sub-interval [2,3]: This rectangle has height f ( 2 ) = 17 and width 1, so its area is 17.

Sub-interval [3,4]:This rectangle has height f (3) = 11 and width 1, so its area is 11.

Adding the areas of these rectangles, we estimate the area between the graph of f and the x-axis on [0,4] to be

20 + 18 + 17 + 11 = 66.

  • Are the estimates in parts (b) and (c) over- or under-estimates for the area between the function f and the x-axis on the interval [0,4]?
    Answer. We don't know what the function f looks like exactly, but we know it's a decreasing function that passes through these points.

That means it must look something like this:

Our estimates in (b) and (c) were both underestimates, because the rectangles didn't cover all of the area between the graph of f and the x-axis on [0,4]:

  • Could we use a right-hand sum with more than 4 sub-intervals to estimate the area between the function f and the x-axis on the interval [0,4]?

Answer. No. The table doesn't contain enough data for us to divide the interval [0,4] into more than 4 sub-intervals. If we tried to use 8 sub-intervals, for example, we would need to know f (0.5), and that value isn't in the table.

Right-Hand Sums with Graphs

When finding a right-hand sum, we need to know the value of the function at the right endpoint of each sub-interval. We can find these values by looking at a graph of the function.

Right-Hand Sum Calculator Shortcuts

For a LHS, we only use values of the function at left endpoints of subintervals. We never use the value of the function at the right-most endpoint of the original interval.

For Right-Hand sums, it's the other way around. For a RHS we only use values of the function at right endpoints, so we'll never use the value of the function at the left-most endpoint of the original interval.

Right-Hand Sums with Math Notation

After learning the notation for left-hand sums, the notation for right-hand sums requires a very slight adjustment. Assume that we're using sub-intervals all of the same length and we want to estimate the area between the graph of f ( x ) and the x-axis on the interval [a,b].

An interval of the form [a,b] has length ( ba ). If we wish to divide the interval [a,b] into n equal sub-intervals, each sub-interval will have length

The endpoints of the sub-intervals are

To take a RHS with n sub-intervals we find the value of f at every endpoint but the first, add these values, and multiply by Δx.

RHS(n) = [f (x1) + f (x2) + ... + f (x n – 1) + f (xn)]Δx

Summation notation for extra fanciness is optional:

Right-Hand Sums with Sub-Intervals on Different Lengths

As with left-hand sums, we can take right-hand sums where the sub-intervals have different lengths.

Sample Problem

Values of the function f are shown in the table below. 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 [1,8].

Answer. The sub-intervals given in this table aren't all the same. Most of them are 2, but one is 1.

On sub-interval [1,3] the height of the rectangle is f (3) = 5 and the width is 2, so the area is

5(2) = 10.

On sub-interval [3,4] the height of the rectangle is f (4) = 3 and the width is 1, so the area is

3(1) = 3.

On sub-interval [4,6] the height of the rectangle is f (6) = 5 and the width is 2, so the area is

5(2) = 10.

On sub-interval [6,8] the height of the rectangle is f (8) = 1 and the width is 2, so the area is

1(2) = 2.

Adding the areas of the rectangles, we estimate the area between f and the x-axis on [1,8] to be

10 + 3 + 10 + 2 = 25.

Example 1

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

Use a right-hand sum with two sub-intervals to approximate the area of R.


Example 2

Let R be the region between the graph y = f ( x ) = x2 + 1 and the x-axis on the interval [0,4]. Use a Right-Hand Sum with 4 sub-intervals to estimate the area of R.


Example 3

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

Use a right-hand sum with 2 sub-intervals to estimate the area of S. Is this an under-estimate or an over-estimate?


Example 4

Let f ( x ) = 2 + x2 and let R be the region between the graph of f and the x-axis on the interval [0,8].

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


Example 5

Let f ( x ) = 4x and let R be the region between the graph of f and the x-axis on the interval [1,2]. Use a right-hand sum with 4 sub-intervals to estimate the area of R.


Example 6

Let f (x) = 2x on [2,10]. Find RHS(5). That is, use a right-hand sum with 5 sub-intervals to estimate the area between the graph of f and the x-axis on [2,10].


Exercise 1

Let R be the region between the graph y = 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?

Exercise 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 (b) and (c) bigger or smaller than the actual area of S?

Exercise 3

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

  • Draw W.\item 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 in parts (b) and (c) bigger or smaller than the actual area of W?

Exercise 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 in (a) and (b) over- or under-estimates of the actual area between the graph of f and the x-axis on the interval [2,8]?

Exercise 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 in (a) and (b) over- or under-estimates for the actual area between the graph of g and the x-axis on the interval [-1,2]?

Exercise 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.

Exercise 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 in (a) and (b) over- or under- estimates for the area of Z?

Exercise 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].

Exercise 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].

Exercise 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]. 

Exercise 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]. 

Exercise 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.

Exercise 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].

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