Introduction to Integral Calculus - Finite Sums

Integral Calculus

Oh no.

- Lightning McQueen

"I am kerchoo."

Introduction

Imagine...

Imagine you're the passenger on a road trip. Every so often, you ask the driver how fast you're going. Can you use this information to figure out how far you traveled, when you finally reach your destination?

Open Ended

How would you try to answer to this question? What information would you need? What equations/relationships/formulae might you use? Take a few minutes and discuss possible approaches in small groups of 2-3.

A Quick Road Trip

Below is the record of speed in miles per hour from this imagined road trip. How can we determine how far we traveled?

​Speed (mph)	​20	65​	75​	15​
​	​0	​60	​120	180​

Speed and Distance

The relationship between speed and distance is a relatively simple one -- if you know the distance between two points, and the amount of time taken to reach one point from another, you can find the average speed.

Speed and Distance

So, if you know how fast an object was going, and the time that it was traveling at that speed, you can figure out how far it went!

Generalizing The Formula

In the case of our road trip, we have four different speeds measured at four different points in time. If we assume the speed did not change from one measurement until the next measurement was taken, then we can get a rough idea of how far the car traveled.

Poll

For the road trip in the example, how far did the car travel?

Hint: Be very careful not to mix your units. If you apply dimensional analysis, you should end up with only units of distance.

• $$160\ \text{miles}$$

• $$155\ \text{miles}$$

• $$9300\ \text{miles}$$

• $$9500\ \text{miles}$$

Answering the Question

First, we need all units to be the same. So we will convert the second row of the table into units of hours.

​Speed (mph)	​20	65​	75​	15​
​	​0	​1	2​	3​

Answering the Question

Now that the units are consistent, we need to see how long elapsed between each measurement. We can do this by finding the difference between two adjacent columns. If we find that the elapsed time is consistent, our lives will be just a bit more easy.

Answering the Question

Open Ended

For our current example, we have four points of data, but only three intervals of time separating each point. Do you think that this is a problem? If so, how would you work past it to find the answer we're looking for?

Assume that each measurement applies to the interval of time up until the measurement was taken.

Right Endpoint Approximation

Assume that each measurement applies to the interval of time up until the next measurement is taken.

Left Endpoint Approximation

Answering the Question

Answering the Question

There is no "Right Choice" here.

Both of these decisions are valid, but will lead us to slightly different answers. Which option we choose depends entirely on the context of the problem at hand. Given this particular situation, I will choose the left endpoint approximation -- but if you chose a right endpoint approximation, that is just fine.

Answering the Question

Now for the actual math.

By assuming that the speed is constant between measurements, we are constructing 3 rectangles. The combined area of these rectangles is the approximate distance traveled.

A Geometric Perspective

Draw

The graph shown on the previous slide corresponds to the left endpoint approximation. Sketch a similar graph showing what you think the right endpoint approximation may look like.

Open Ended

Both the left and right endpoint approximations give similar answers to our road trip example. Why do you think this is? Can you think of at least one way that we can improve on both of these approximations?

Improving our Answer...

The left and right endpoint approximations give similar results when evaluated. But which one is closer to reality? There are a few ways that we could improve our answer here, but we will focus on just one for now.

Left Endpoint Approximation

The Midpoint Approximation averages out the left and right endpoint approximations, to give a mean value in between. We can calculate this by first finding the mean value of each interval, and using that value to calculate the height of each rectangle!

Midpoint Approximation

Right Endpoint Approximation

The midpoint approximation can be calculated by first finding the midpoint of each subinterval. For our specific example, since we have so few data points, this means that we can imagine a line segment connecting each data point. The midpoint of each line segment will be the height of each rectangle. In general, though, it can be a bit more complicated.

Midpoint Approximation

A More Accurate Answer

By assuming that there is a straight line connecting each data point, we simplify the math slightly; We can simply calculate the midpoint of each line segment:

​In general, it's not quite this simple -- but we mustn't get ahead of ourselves.

A More Accurate Answer

Now for the actual math. Again.

Word Cloud

What are some other real-world situations where you think this technique may be useful?

Generalizing the Formulae

These Can Work Almost Everywhere!

These approximations that we just discussed can be generalized to work with almost any function! As long as there aren't any:

• Vertical Asymptotes

• Discontinuities

• or Rapid Oscillations!

​In these situations, we turn to other techniques, which we will see later!

Almost Done...

Almost Done...

Wait... Closed Subintervals??

An interval is a set of numbers which contains every number between two endpoints, without any "gaps." If both endpoints of the interval are included within, then the interval is "closed." A "closed subinterval" is an interval which we construct to contain some, but not necessarily all, of the numbers in some larger interval.

Yes! Closed Subintervals - Math :)

Almost Done...

Almost Done...

If we wish to find the left endpoint approximation, then we want the height of each rectangle to be calculated using the left endpoint of each subinterval. This corresponds to...

Almost Done...

Almost Done... (No, Really!)

If the area of one rectangle spanning a single subinterval is...

Left Endpoint Approximation

Left Endpoint Approximation

Right Endpoint Approximation

Right Endpoint Approximation

Midpoint Approximation

Midpoint Approximation

Recall from Precalculus...

Practice Time

Match

Match the names with the corresponding equation.

• Left Endpoint Approximation

• Midpoint Approximation

• Right Endpoint Approximation

• $$\sum_{\ k=1}^nf\left(c_{k-1}\right)\cdot\Delta x$$

• $$\sum_{\ k=1}^nf\left(\frac{c_{k-1}+c_k}{2}\right)\cdot\Delta x$$

• $$\sum_{\ k=1}^nf\left(c_k\right)\cdot\Delta x$$

Dropdown

When approximating the area under the curve of a function, Δx represents the ​

Select answer

of a ​

Select answer

Multiple Choice

Approximate the area under the curve of the function $$f\left(x\right)=x$$ , over the interval $$\left[0,4\right].$$ Use the left endpoint approximation, with $$n=4$$ subintervals.

• 8
• 4
• 10
• 6

Multiple Choice

Approximate the area under the curve of the function $$f\left(x\right)=x^2$$ , over the interval $$\left[0,4\right].$$ Use the midpoint approximation, with $$n=4$$ subintervals.

• 21
• 25
• 16
• 12

Open Ended

Is it possible for us to use the any of the approximations to approximate the area under the curve of the function $$f\left(x\right)=\frac{1}{x}$$ on the interval $$\left[0,1\right]$$ ? Why, or why not?

Poll

"I think calculus is..."

• Pretty Neato.

• Radical.

• A form of Psychological Warfare.

• Worse than stubbing my toe.

Poll

On a scale of 1-5, rate the quality of this slideshow and example problem. Do you feel like it was a helpful explanation and introduction?

• 1 - Poor Quality

  I did not find this particularly helpful or informative.

• 2 - Below Average

  The slideshow was helpful, but could be improved greatly.

• 
  3 - Average

  Not great, not horrible.

• 4 - Above Average

  The slideshow and example has room for improvement, but was overall helpful.

• 5 - Excellent

  I found the slideshow and example problem to be a very helpful introduction to the topic.

Open Ended

What, if anything, do you feel would improve the quality of these slides, and/or make the information more accessible to yourself, or others?