Riemann Sums, Summation Notation, and Definite Integral Notation

#6.3 Riemann Sums, Summation Notation, and Definite Integral Notation

So far in Unit 6, we’ve learned how to approximate the area under the curve. But what if we wanted to find its exact value? For that, we’ll need to learn about integrals and how they are related to the Riemann sum. We’re bringing limits back! 🧠

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#Σ Summation Notation

Before we get into integrals, we first need to learn how to make taking a Riemann sum faster. To do this, we’ll introduce summation notation so that we can apply algebra instead of manually computing the Riemann sum.

Let’s consider the function $f(x)=x^2$ over the interval 0 to 5:

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Desmos graph of f(x) = x^2 with area under the curve from [0,5] shaded.

Image courtesy of Emery

Now, let’s use a left Riemann sum with 5 subintervals to approximate this area.

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The same Desmos graph with 5 left-endpoint Riemann rectangles.

Image courtesy of Emery

To convert this to summation notation, we need to create a function that will give us the area of each rectangle. For now, let’s name this function $A$ so that $A(i)$ gives us the area of the $i$th rectangle. With this function, we can write the Riemann sum as:

$$\sum_{i=0}^{4}A(i)$$

Now that we have our summation notation, we need to find an expression for $A(i)$. We know that the area of the rectangle will be $b\cdot h$, where $b$ is the width of the base and $h$ is the height.

The width of our base will be constant, and we can find it by dividing the entire interval by the number of subintervals we want:

$$\frac{5-0}{5}=1$$

The height is a little more tricky to notate, but it is just the value of the function $f$ at each of the left endpoints. To find this next (or $i$th) point, let’s first find the $x$-value, which we’ll denote as $x_i$. To do this, we start at 0 and repeatedly add 1, the length of our subinterval.

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A graph demonstrating the relationship between the length of the subinterval and x_i.

Image courtesy of Emery

We can write the formula for this $x$-value as 1i or just $i$. We can obtain the $y$-value, or height, by simply plugging in this formula for the $x$-value to our original formula like so: $y_i=(i)^2$.

We can put these two things together in the area formula as:

$$A(i)=1\cdot i^2$$

which simplifies to

$$A(i)=i^2$$

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