# Definite Integrals

### Topics

## 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 ( *b* – *a* ). 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 Δ*x*.

*RHS*(*n*) = [*f* (*x*_{1}) + *f* (*x*_{2}) + ... + *f* (*x*_{ n – 1}) + *f* (*x*_{n})]Δ*x*

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

#### Example 2

Let |

#### Example 3

Let Use a right-hand sum with 2 sub-intervals to estimate the area of |

#### Example 4

Let Use a right-hand sum with 4 sub-intervals to estimate the area of |

#### Example 5

Let |

#### Example 6

Let |

#### Exercise 1

Let *R* be the region between the graph *y* = *f** *(* x *) =

*x*+ 1 and the x-axis on the interval [0,4].

^{2}- 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* ) = 2^{x} 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* ) = *x*^{2} + 6*x* + 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* ) = -*x*^{2} + 2*x* + 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].