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  1. AP Calculus
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Riemann Sums, Summation Notation, and Definite Integral Notation

Samuel Baker

Samuel Baker

9 min read

Next Topic - The Fundamental Theorem of Calculus and Accumulation Functions

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Study Guide Overview

This study guide covers Riemann sums, summation notation, and definite integral notation. It explains how to express Riemann sums using summation notation, including the general form for left and right endpoints. It also connects Riemann sums to definite integrals using limits, providing a walkthrough example of converting from summation to definite integral notation. Finally, it offers practice problems and solutions for expressing definite integrals as limits of Riemann sums.

#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! 🧠


#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)=x2f(x)=x^2f(x)=x2 over the interval 0 to 5:

!Untitled

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.

!Untitled

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 AAA so that A(i)A(i)A(i) gives us the area of the iiith rectangle. With this function, we can write the Riemann sum as:

∑i=04A(i)\sum_{i=0}^{4}A(i)i=0∑4​A(i)

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

5−05=1\frac{5-0}{5}=155−0​=1

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

!PNG image 23.png

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 xxx-value as 1i or just iii. We can obtain the yyy-value, or height, by simply plugging in this formula for the xxx-value to our original formula like so: yi=(i)2y_i=(i)^2yi​=(i)2.

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

A(i)=1⋅i2A(i)=1\cdot i^2A(i)=1⋅i2

which simplifies to

A(i)=i2A(i)=i^2A(i)=i2

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Previous Topic - Approximating Areas with Riemann SumsNext Topic - The Fundamental Theorem of Calculus and Accumulation Functions

Question 1 of 8

🎉 What does the symbol ∑\sum∑ signify in the context of Riemann sums?

The product of the areas of all rectangles

The average of the areas of all rectangles

The sum of the areas of all rectangles

The difference between the areas of the largest and smallest rectangles