Area Under the Curve We want to approximate the area between a curve (y=x 2 +1) and the x-axis from x=0 to x=7 We want to approximate the area between a curve (y=x 2 +1) and the x-axis from x=0 to x=7 We will use rectangles to do this. We will use rectangles to do this. One way will be to choose rectangles whose heights are taken from the x-coordinate of the right side of the rectangle (Right Sum) One way will be to choose rectangles whose heights are taken from the x-coordinate of the right side of the rectangle (Right Sum) We will let n be the number of rectangles we use to approximate the area We will let n be the number of rectangles we use to approximate the area

Right Sum n=5

Right Sum n=10

Right Sum n=20

Right Sum n=50

Right Sum n=100

What do you notice? What will happen as you add more rectangles for a Right Sum? What will happen as you add more rectangles for a Right Sum? Will this happen for any function? Why or why not? Will this happen for any function? Why or why not? How many rectangles do we need to get the actual area? How many rectangles do we need to get the actual area? Can you think of another way to approximate the area? Can you think of another way to approximate the area?

Another way: What if we were to use rectangles whose heights were formed from the x-coordinate of the Left side of the interval? What if we were to use rectangles whose heights were formed from the x-coordinate of the Left side of the interval? We will call these Left Sums We will call these Left Sums As we go through the Left Sums, what do you notice about the areas? As we go through the Left Sums, what do you notice about the areas?

Left Sum n=5

Left sum n=10

Left Sum n=20

Left Sum n=50

Left Sum n=100

Let’s look at the areas again: Right Sums Left Sums n= n= n= n= n= n= n= n=

Another Example While it is easiest for computational reasons to look at Left Sums and Right Sums, theoretically it is necessary to look at Upper Sums (where each rectangle circumscribes the function) and Lower Sums (where each rectangle is inscribed in the function). Recall that in the example seen so far, the Right Sums were also Upper Sums and the Left Sums were Lower Sums. Under what circumstances would this not be true? Is it possible for some Upper Sums to be left sums and others to be right sums? While it is easiest for computational reasons to look at Left Sums and Right Sums, theoretically it is necessary to look at Upper Sums (where each rectangle circumscribes the function) and Lower Sums (where each rectangle is inscribed in the function). Recall that in the example seen so far, the Right Sums were also Upper Sums and the Left Sums were Lower Sums. Under what circumstances would this not be true? Is it possible for some Upper Sums to be left sums and others to be right sums?

Here is an example of an Upper Sum. Notice that the rectangles are all formed by choosing the height from the highest point on the graph that the rectangle hits.

Here is a Lower Sum with 8 rectangles. What do you think will happen if a rectangle starts at -½ and ends at ½? What would the height be for a Lower Sum?

Finding Exact Areas Riemann Sums It turns out, as you could see from the table, that if you use enough rectangles, the Left Sum will be very close to the Right Sum. If you use an infinite amount of rectangles, the Left Sum and Right Sum will be equal, and they will equal the exact area. It turns out, as you could see from the table, that if you use enough rectangles, the Left Sum will be very close to the Right Sum. If you use an infinite amount of rectangles, the Left Sum and Right Sum will be equal, and they will equal the exact area. Theoretically, there are a great many ways to design rectangles that will lead to the area. Any of these will be a type of Riemann Sum. So far we have seen Left, Right, Upper, and Lower Sums; these are all types of Riemann Sums. There are lots of others. Theoretically, there are a great many ways to design rectangles that will lead to the area. Any of these will be a type of Riemann Sum. So far we have seen Left, Right, Upper, and Lower Sums; these are all types of Riemann Sums. There are lots of others. Since any Riemann Sum will eventually lead to the exact area, we will use the Right Sum. The Right Sum is computationally easiest to use. Since any Riemann Sum will eventually lead to the exact area, we will use the Right Sum. The Right Sum is computationally easiest to use.

Finding Exact Areas Riemann Sums Back to Back to If we use 8 rectangles, how wide is each one? If we use 8 rectangles, how wide is each one? What about if we used 12 rectangles? What about if we used 12 rectangles? How about n rectangles? How about n rectangles? In general, if we start at x=a and stop at x=b, how wide would each rectangle be? In general, if we start at x=a and stop at x=b, how wide would each rectangle be?

Exact Area In general, the width of each rectangle will be In general, the width of each rectangle will be We call the width of each rectangle We call the width of each rectangle What would the height of each rectangle be? Let’s look at the graph again, and then see if we can generalize… What would the height of each rectangle be? Let’s look at the graph again, and then see if we can generalize…

As you can see, the heights of each rectangle are found by getting the y-value at the right side of each rectangle. If x i represents the x-coordinate of the right side of the i th rectangle, then the height of the i th rectangle is f( x i ). For our example, each x i is found by adding to the previous right side. Since we start at x=0, the first right side is the next one is. Without doing the addition, what would be the coordinate of the right side of the 5 th rectangle? Do you see that you merely multiply 5 times ? How would this generalize?

If the area we are interested starts at x=a, and the width is then the x-coordinate of the i th rectangle x i will be If the area we are interested starts at x=a, and the width is then the x-coordinate of the i th rectangle x i will be Example: If I was finding the area under a curve on the interval [3, 7] and I was using 100 rectangles, the x-coordinate of the 70 th rectangle would be Example: If I was finding the area under a curve on the interval [3, 7] and I was using 100 rectangles, the x-coordinate of the 70 th rectangle would be Find x 25 for an area on the interval [2, 8] if we use 120 rectangles Find x 25 for an area on the interval [2, 8] if we use 120 rectangles Did you get 3.25? Did you get 3.25?

Area of the i th rectangle Since area of a rectangle is height times width, and the height is just the value of the function at x i, we get Since area of a rectangle is height times width, and the height is just the value of the function at x i, we get Given on the interval [5, 7] with 20 rectangles, find the area of the 14 th rectangle. Given on the interval [5, 7] with 20 rectangles, find the area of the 14 th rectangle. Did you get Did you get

Putting it all together Now we want to put it all together. We want to add up all n rectangles to give an approximation for the area. This is the formula we use: Now we want to put it all together. We want to add up all n rectangles to give an approximation for the area. This is the formula we use: Let’s go back to our first example: on [0, 7] and let’s use 100 rectangles. We get Let’s go back to our first example: on [0, 7] and let’s use 100 rectangles. We get which is what we had before on our table.

The exact area To find the exact area, all we need to do is look at an infinite number of rectangles. Believe it or not, this is actually easier than what we just did. The formula becomes To find the exact area, all we need to do is look at an infinite number of rectangles. Believe it or not, this is actually easier than what we just did. The formula becomes The previous example becomes The previous example becomes Remember that the sum of a constant is the constant times n. Remember that the sum of a constant is the constant times n.

Another example Find the area under on the interval [2, 5] Find the area under on the interval [2, 5] We have, and We have, and So the area is So the area is

A few last comments Don’t Panic – an easier way is coming soon Don’t Panic – an easier way is coming soon There are many, many applications of what we just did…also coming soon There are many, many applications of what we just did…also coming soon What do you think would change if we tried to find the exact area using a Left Sum? Why is using a Left Sum more complicated? What do you think would change if we tried to find the exact area using a Left Sum? Why is using a Left Sum more complicated? Another, very common method used to approximate the area under a curve is called a Midpoint Sum. Without any other information, what do you think that might be? Another, very common method used to approximate the area under a curve is called a Midpoint Sum. Without any other information, what do you think that might be? To approximate the area under a curve, we usually just use a few rectangles, and which method we use depends on what the graph looks like. To get the exact area we use Right Sums, but we are limited to only finding the area under polynomial of degree 3 or less (Why?) To approximate the area under a curve, we usually just use a few rectangles, and which method we use depends on what the graph looks like. To get the exact area we use Right Sums, but we are limited to only finding the area under polynomial of degree 3 or less (Why?)

A final comment or three If you are approximating the area under a curve and you are using only a few rectangles it is MUCH easier to find the heights and areas by hand (or using a table in your calculator) than to use formulas. If you are approximating the area under a curve and you are using only a few rectangles it is MUCH easier to find the heights and areas by hand (or using a table in your calculator) than to use formulas. The width of each rectangle will not always be the same…think about how you approach that situation. The width of each rectangle will not always be the same…think about how you approach that situation. On an AP test you will need to do Left, Right, Upper, Lower, and/or Midpoint Sums for small numbers of rectangles. On an AP test you will need to do Left, Right, Upper, Lower, and/or Midpoint Sums for small numbers of rectangles.