Finding Areas with Riemann Sums |

**Warning:** You must know summation notation, formulae, and operations do this lesson.

**Area Under a Curve **

In order to find the area between a given function's curve and the *x-axis*, or between two given curves, mathematical pioneers decided to divide the desired area into a finite number of rectangles with equal or unequal bases and then sum the areas of these rectangles.

This method yields a close estimate of the desired area however, it also includes or excludes spaces, since the rectangles don't exactly fit the area defined by the curve. There are always little bits more or less than the area we want.

They solved this problem by **limiting n** -- the number of rectangles --

So we'll deal with two types of questions on finding area from fundamental principles. In the first type, we'll divide the area up into rectangles with **equal bases** and in the second type, we'll make a **rectangular partition** of the area with **predefined borders**. In the **first case**, we'll get the exact area we want because we will **limit** ** n** -- the number of rectangles --

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**Riemann Sums**

Say we want to find the area under the curve** f (x) = x² **from

We divide the 5 units of the x-axis into

Now we need to find the

We see that the right endpoints of each rectangle's base is:

in rectangle #1,

in rectangle #2,

so in the *" i" *

where

Since , , the

The area of a rectangle is height times base, so the area of the ** i th** rectangle here is:

If we sum the above expression over the interval from 0 to 5, we will have the area.

The problem however, is that we find not only the area under the curve, but also the area of the triangles above the curve. To eliminate these, we **limit n, the number of rectangles, to infinity**. This makes the bases teeny since there are so many of them and so gives us the area between the

curve ** f (x) = x^{2}** and the

So our final expression for the area we're seeking is:

We have a formula for the summation of *i*^{ 2} , we apply it now and then take the limit as n approaches infinity. Recall, to do this we divide our fraction through by the highest power of the variable, then set the variable equal to infinity.

When we switch the denominators and find the limit for the 2nd fraction (with all the ** n**'s in it),

we get 2, since the highest power of

When we reduce the fraction, the answer is .

Now say we wish to find the area between the ** x-axis** and the curve

between

The process is almost identical except now, , and ,

so

When we multiply by *f *(*x*) we get:

So our area this time is:

When we substitute the formulae for the summations of ** i** and

Now let's do it by integration. We evaluate

**Note** that the area is a summation of the __product__ ** f (x_{ i })** times which is

**Notation**

When dealing with partitions, there are a number of notation systems used to denote the

values of *x* that define the borders and heights for the rectangles.

Generally, the border values for the partitions are denoted *x _{ i} *,

Some texts use *x*_{*}* *for the ** x-value** used to find the height or

**Riemann Sums with Partitions**

Now let's evaluate a **partition** of the area under the curve *f* (*x*) = *x***² + 1** on the interval from

*a*** = 0 ** to *b*** = 6** and with *n*** = 5** and *x _{ }*

We'll use the

As we see from the ** x-values**, the length of the 1st four bases = 1, the 5th = 2.

The midpoint values are:

w = 0.5_{1} |
w = 1.5_{2} |
w = 2.5_{3} |
w = 3.5_{4} |
w = 5_{5} |

We need values for *f*(*x _{ i }*) since those are the heights of the rectangles.

f (w) = 1.25_{1} |
f (w) = 3.25_{2} |
f (w ) = 7.25_{3} |
f (w ) = 13.25_{4} |
f (w) = 26_{5} |

The sum = 1.25 + 3.25 + 7.25 + 13.25 + 2(26) = 77 (units) ² , so this our estimate for the area.

Though it is not obvious in the diagram, we're also finding the area of the little triangles above the curve ** y = x^{ }2 + 1**. To eliminate this extra area, Riemann stated that we need only limit the largest base to zero in order to limit all bases to zero.

In our case however, we need only estimate the Riemann sum described in the question.

There are times when we must use the left or right endpoint of the base interval. The question will say which one to use. If there are no specific instructions, we should always use the right endpoint since it's easiest to find.

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Should the curve in question cross the *x-axis* at some point on the given interval, we must take into account that the *heights* of the rectangles below the axis will all be negative. Since area can't be negative, we break it up into two parts. We multiply all negative heights by – 1 to make them positive, we find the two areas separately and then sum them. For this reason, it's always best to make a diagram of the situation in question and to check the interval for intercepts.

Here, we'll find the area under *f *(*x*) from *x* =0 to *x* = 5 and we'll add it to the area under

– *g*(*x*) from *x* = 5 to *x* = 7, since multiplying the function *g*(*x*)** **by – 1 flips the curve

to generate the identical area above the

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**Summary**

To find the exact **area under a curve** *f *(*x*):

1. Divide the *x-axis* interval (*a, b*) into *n* rectangles with **equal bases.**

2. Find an expression for , the length of each base, .

3. Find an expression for ** x_{ i}** , (with

4. Find * *an expression for ** f ( x_{ i}** ) , the height of the

5. Multiply height ** f ( x_{ i}** ) by base , to get an expression for the area of

6. Sum the formula in step 5 from *i = 1* to *n*. Substitute the summation formulae.

7. Limit *n* to ** infinity** to eliminate the under- or over-hang area.

To **estimate area** under a curve using a **Partition**:

1. Draw a diagram using the border values defined in the question.

2. Make a table of values for , ** x_{ i}** , and

3. Multiply base by height for each rectangle then sum the areas.

1) Use a Riemann sum to find the area between *f (x)* and the *x*-axis, over the given interval.

a) f (x) = 5 – x²with |
b) f (x) = x³ + 8with |
c) f (x) = x² + 1with |

| (summation formulae) | (solutions)

2) Estimate the area under the curve *f* (*x*)** = x² + 1** from

Use the

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1) a)

We see that , so which makes the **height**

The area of the ** i th** rectangle is

The total area then is

Now, as before, we switch the denominators and evaluate the summation of *i*².

So we get square units.

When we integrate the function from 0 to 2 we get exactly the same value for the area.

b)

We see that , so which makes the **height**

The area of the ** i th** rectangle is

The total area then is

Now, as before, we switch the denominators and evaluate the summation of *i*³.

So we get square units.

When we integrate the function from 0 to 5 we get exactly the same value for the area.

c)

We see that , but since we start at 1,

which makes the **height** ** f** (

The area of the ** i th** rectangle is

The area then is

Now, as before, we switch the denominators and evaluate the summation of *i* and *i*².

So we get square units.

When we integrate the function from 1 to 3 we get exactly the same value for the area.

2) use diagram above

The length of each base = 1

The midpoint values are:

w = 0.5_{1} |
w = 1.5_{2} |
w = 2.5_{3} |
w = 3.5_{4} |
w = 4.5_{5} |
w = 5.5_{6} |

We need values for *f *(*x _{ i }*) since those are the heights of the rectangles.

f (w) = 1.25_{1} |
f (w) = 3.25_{2} |
f (w ) = 7.25_{3} |
f (w ) = 13.25_{4} |
f (w) = 21.25_{5} |
f (w) = 30.25_{6} |

The total = 1.25 + 3.25 + 7.25 + 13.25 + 21.25 + 30.25 = 76.5 units ² , so this our estimate for the area.

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a) | b) |

c) | d) |

e) | f) ^{ } |

g) |

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