Definite Integrals
Topics
Introduction to Definite Integrals  At A Glance:
We have formulas to find areas of shapes like rectangles, triangles, and circles (pi, anyone?).
What if we want to find the area of a lessreasonable shape? Think of sea monkeys. Sure, we all want them. We want to race them and bring them along on space adventures—but how many can we fit in our twodimensional space ship? We have to use integrals to figure out the area of each sea monkey before heading into orbit.
Let R be the region between the graph y = f ( x ) = x^{2} + 1 and the xaxis on the interval [0,4]:
R is a weird shape, and we don't have any formula that says how to find its area (yet!).
In this section we're going to look at ways to approximate the areas of shapes that are formed, like R, by graphing nonnegative functions on specified intervals.
A nonnegative function is as it sounds: a function that never outputs a negative yvalue.
Sample Problem
The following function is nonnegative (hitting zero is allowed):
Sample Problem
The following function is not nonnegative:
Back to R—we can approximate the area of R by drawing rectangles that moreorless cover R, and calculating the total area covered by those rectangles.
There are several different procedures for drawing these rectangles. The most important ones to know are the LeftHand Sum, the RightHand Sum, and the Midpoint Sum.
These are examples of Riemann Sums. There's also a procedure called the Trapezoid Sum, which draws trapezoids instead of rectangles.
The first step in any of these procedures is to chop up the original interval into subintervals, usually all of the same size.
On each subinterval we draw a rectangle whose base is that subinterval. The height of each rectangle depends on which procedure we're using.
With a LeftHand Sum (LHS) the height of the rectangle on a subinterval is the value of the function at the left endpoint of that subinterval.
We can find the values of the function we need using formulas, tables, or graphs.
When finding a lefthand sum, we need to know the value of the function at the left endpoint of each subinterval. One way to find these function values is to calculate them using a formula for the function.
LeftHand Sums with Formulas
When finding a lefthand sum, we need to know the value of the function at the left endpoint of each subinterval. One way to find these function values is to calculate them using a formula for the function.
LeftHand Sums with Tables
In order to find a lefthand sum we need to know the value of the function at the left endpoint of each subinterval. We can take a lefthand 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 LeftHand Sum with 2 subintervals to estimate the area between the graph of f and the xaxis on the interval [0,4].
Answer. 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 equallysized subintervals gives us subintervals of length 2.
The height of the rectangle on [0,2] is f ( 0 ) = 20, so the area of this rectangle is
height ⋅ width = 20(2) = 40.
The height of the rectangle on [2,4] is f ( 2 ) = 17, so the area of this rectangle is
height ⋅ width = 17(2) = 34.
Adding the areas of these rectangles, we estimate the area between the graph of f and the xaxis on [0,4] to be
40 + 34 = 74.
 Use a LeftHand Sum with 4 subintervals to estimate the area between the graph of f and the xaxis on the interval [0,4].
Answer. Dividing the interval [0,4] into 4 evenlysized subintervals produces subintervals of length 1:
Subinterval [0,1]: This rectangle has height f ( 0 ) = 20 and width 1, so its area is 20.
Subinterval [1,2]: This rectangle has height f (1) = 18 and width 1, so its area is 18.
Subinterval [2,3]: This rectangle has height f (2) = 17 and width 1, so its area is 17.
Subinterval [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 xaxis on [0,4] to be
20 + 18 + 17 + 11 = 66.
 Are the estimates in parts (b) and (c) over or underestimates for the area between the function f and the xaxis 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 dots:
That means it must look something like this:
and so our estimates in (b) and (c) were both overestimates, because the rectangles covered extra area:
 Could we use the lefthand sum with more than 4 subintervals to estimate the area between the function f and the xaxis 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 subintervals. If we tried to use 8 subintervals, for example, we would need to know f (0.5), and that value isn't in the table.
LeftHand Sum with Graphs
When finding a lefthand sum, we need to know the value of the function at the left endpoint of each subinterval. We can find these values by looking at a graph of the function.
LeftHand Sum Calculator Shortcuts
When working through lefthand sums, we need to multiply every function value we use by the width of the rectangle. We can use these observations to work more efficiently and make better use of the calculator. Check out the examples for more ideas.
LeftHand Sum with Math Notation
It's possible to write the process for taking a LHS as a nice tidy formula. Assume that we're using subintervals all of the same length and we want to estimate the area between the graph of f ( x ) and the xaxis on the interval [a,b].
An interval of the form [a,b] has length ( b – a ). This is true whether or not the numbers a and b are positive or negative. For example, the interval [1,10] has length 11.
If we wish to divide the interval [a,b] into n equal subintervals, each subinterval will have length
This quantity is often called Δ x:
We split the interval [a,b] into n subintervals, each of length . We need to know the value of f at each endpoint except at x = b (the rightmost endpoint of the original interval).
We need to know what those endpoints are. The first one is x = a. The next endpoint is a + Δx. The next is (a + Δx) + Δx = (a + 2Δx). Then (a + 2Δx) + Δx = (a + 3Δx).
When do we stop? We know that
This means if we start at a and take n steps of size Δx, we'll get to b, which is the end of the original interval. The last endpoint we need for a lefthand sum is the one just before b, which is
a + ( n – 1 )Δx
These endpoints are often labeled like this, so we don't need to write down as much:
To take a LHS we find the value of f at every endpoint but the last, add these values, and multiply by the width of a subinterval. In this case, the width of a subinterval is Δx.
[f (x_{0}) + f (x_{1}) + f (x_{2}) + ... + f (x_{ n – 1 })](width) = [f (x_{0}) + f (x_{1}) + f (x_{2}) + ... + f (x_{ n – 1 })]Δx
Using a lefthand sum with n subintervals, we estimate the area between the graph of f and the xaxis on the interval [a,b] is
LHS(n) = [f (x_{0}) + f (x_{1}) + f (x_{2}) + ... + f (x_{ n – 1 })]Δx.
If we wanted to be extra fancy, we could use summation notation. Using i to keep track of which endpoint we're on, we can write the lefthand sum as
.
This formula is the same thing as the calculator shortcut. It's a short, tidy way to write down the process for taking a lefthand sum. There are two important things to remember.
 For a lefthand sum, the last endpoint you use is x_{ n – 1 }:
 After adding up the values of f at the appropriate endpoints, multiply by the width of a subinterval:
LeftHand Sum with SubIntervals of Different Lengths
All the lefthand sums we've found so far had subintervals that were all the same length. It doesn't need to be this way. There are some situations where we want to use subintervals of different lengths.
Sample Problem
Values of the function f are shown in the table below. Use a lefthand sum with the subintervals indicated by the data in the table to estimate the area between the graph of f and the xaxis on the interval [1,8].
Answer. The subintervals given in this table aren't all the same. Most of them are 2, but one is 1.
On subinterval [1,3] the height of the rectangle is f (1) = 2 and the width is 2, so the area is
2(2) = 4.
On subinterval [3,4] the height of the rectangle is f (3) = 5 and the width is 1, so the area is
5(1) = 5.
On subinterval [4,6] the height of the rectangle is f (4) = 3 and the width is 2, so the area is
3(2) = 6.
On subinterval [6,8] the height of the rectangle is f (6) = 5 and the width is 2, so the area is
5(2) = 10.
Adding the areas of the rectangles, we estimate the area between f and the xaxis on [1,8] to be
4 + 5 + 6 + 10 = 25.
Example 1
Let R be the region between the graph y = f ( x ) = x^{2} + 1 and the xaxis on the interval [0,4]: Use a lefthand sum with two subintervals to approximate the area of R. 
Example 2
Let R be the region between the graph y = f ( x ) = x^{2} + 1 and the xaxis on the interval [0,4]. Use a LeftHand Sum with 4 subintervals to estimate the area of R. 
Example 3
Let S be the region between the graph of g and the xaxis on the interval [0,4]. Use a lefthand sum with 2 subintervals to estimate the area of S. Is this an underestimate or an overestimate? 
Example 4
Use a lefthand sum with 4 subintervals to estimate the area of W.

Example 5
Let f ( x ) = x^{2} + 2 and let R be the region between the graph of f and the xaxis on the interval [0,8]. Use a lefthand sum with 4 subintervals to estimate the area of R. 
Example 6
Let f ( x ) = 4x and let R be the region between the graph of f and the xaxis on the interval [1,2]. Use a lefthand sum with 4 subintervals to estimate the area of R. 
Example 7
Let f ( x ) = 2x on [2,10]. Find LHS(5). That is, use a lefthand sum with 5 subintervals to estimate the area between the graph of f and the xaxis on [2,10]. 
Example 8
Use a lefthand sum to estimate the area between the graph of g and the xaxis on the interval [0,10]. 
Exercise 1
Let R be the area between the graph of f ( x ) = x^{2} + 1 and the xaxis on the interval [0,4].
 Draw R and the 8 rectangles that result from using a LeftHand Sum with 8 subintervals to approximate the area of R.
 Use the LeftHand sum with 8 subintervals to approximate the area of R (you might want a calculator).
Exercise 2
 Let S be the area between the graph of y = f ( x ) = 2^{x} and the xaxis on the interval [1,6].
 Draw S.
 Use a LeftHand Sum with 2 subintervals to approximate the area of S. Draw S and the rectangles used in this LeftHand Sum on the same graph.
 Use a LeftHand Sum with 5 subintervals to approximate the area of S. Draw S and the rectangles used in this LeftHand 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 xaxis on the interval [1,4].
 Draw W.
 Use a LeftHand Sum with 3 subintervals to approximate the area of W. Draw W and the rectangles used in this LeftHand Sum on the same graph.
 Use a LeftHand Sum with 6 subintervals to approximate the area of W. Draw W and the rectangles used in this LeftHand 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 lefthand sum with one subinterval to estimate the area between the graph of f and the xaxis on the interval [2,8].
 Use a lefthand sum with three subintervals to estimate the area between the graph of f and the xaxis on the interval [2,8].
 Are your answers in (a) and (b) over or underestimates of the actual area between the graph of f and the xaxis on the interval [2,8]?
Exercise 5
 Some values of the decreasing function g are given in the table below:
 Use a lefthand sum with 3 subintervals to estimate the area between the graph of g and the xaxis on the interval [1,2].
 Use a lefthand sum with 2 subintervals to estimate the area between the graph of g and the xaxis on the interval [1,2].
 Are your answers in (a) and (b) over or underestimates for the actual area between the graph of g and the xaxis on the interval [1,2]?
Exercise 6
 Let f ( x ) = x^{2} + 6x + 9. Use a lefthand sum with 6 subintervals to estimate the area between the graph of f and the xaxis on the interval [6,3].
Exercise 7
 Let f ( x ) = x^{2} + 2x + 8. Use a lefthand sum with 8 subintervals to estimate the area between the graph of f and the xaxis on the interval [0,4].
Exercise 8
Let g be a function with values given by the table below.
Use a lefthand sum with 3 subintervals to estimate the area between the graph of g and the xaxis on the interval [0,12].
Exercise 9
Let h be a function with values given by the table below. Use a lefthand sum with 9 subintervals to estimate the area between the graph of h and the xaxis on the interval [9,9].
Exercise 10
The function f ( x ) on the interval [0,30] is graphed below. Use a lefthand sum with 3 subintervals to estimate the area between the graph of f and the xaxis on this interval.
Exercise 11
Use a lefthand sum with the subintervals indicated by the data in the table to estimate the area between the graph of f and the xaxis on the interval [10,1].
Exercise 12
Use a lefthand sum with the subintervals indicated by the data in the table to estimate the area between the graph of g and the xaxis on the interval [0,5π].
Exercise 13
Use a lefthand sum to estimate the area between the graph of h and the xaxis on the interval [2,7].