Definite Integrals

Definite Integrals

Like, zoinks, Scoob. They're like, adding a bunch of rectangles together.

"the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours,"

- H.P. Lovecraft

Previously...

Area Approximations

Last time, we saw three different ways that we could approximate the area under the curve of a function. These approximations relied on constructing a finite number of rectangles and adding together their areas.

Riemann Sums

The Definite Integral

We will now work to construct the Definite Integral. The goal is to be able to go from merely approximating the area under the curve of some functions, to being able to find the exact area under the curve of a function for any interval on which the function is defined and well-behaved.

The Partition

The Partition

The Partition

The Partition

The Darboux Sums

Upper Darboux Sum

Lower Darboux Sum

The Darboux Sums

On any subinterval, the upper Darboux sum is determined by the local maximum of the function on the subinterval.

Upper Darboux Sum

On any subinterval, the lower Darboux sum is determined by the local minimum of the function on the subinterval.

Lower Darboux Sum

The Darboux Sums

Upper Darboux Sum

Lower Darboux Sum

Lower Darboux Sum

Upper Darboux Sum

The upper and lower Darboux sums overlayed ontop of one another show clearly the difference between the two sums. This is shown here with diagonal shading.

The Darboux Sums

As the number of subintervals is increased, the difference between the upper and lower Darboux sums becomes smaller. If this happens, we say that they converge.

The Darboux Sums

The Darboux Sums

The Definite Integral

The Definite Integral

The Definite Integral

A Bit of Notation

When Do Integrals Exist?

The Definite Integral of a Continuous Function

The definition of the definite integral from previous slides can be simplified if we partition all subintervals to be of equal width.

The Definite Integral of a Continuous Function

The Definite Integral of a Continuous Function

Finally!

At last, we have arrived at the definition of the definite integral. Kind of important for Integral Calculus, don't you think? It seems like a lot, but it won't be too bad after some example problems and practice. Before we get to that though, we should break down the notation used for definite integrals.

The integral of a function can be broken down into the five pieces above. When the value of an integral is determined, we say that the integral has been evaluated.

Let's Break it Down

Example Time!

Now you will see how we can work one of these problems from start to finish.

Example 1

Example 1

Example 1

Example 1

Example 1

Example 1

Example 1

Example 1

Successive Differences

Here, we had to repeat the process twice. The first iteration of successive differences resulted in the second row (labeled 1). The second iteration resulted in the last row (labeled 2).

Successive Differences

Next, we subtract every two adjacent terms, and write down the difference. We repeat this until we find that the difference between two consecutive terms is constant.

Successive Differences

Successive Differences

Multiple Choice

What is the solution to the given system of equations?

$$A+B+C=1$$

$$4A+2B+C=3$$

$$9A+3B+C=6$$

• $$A=\frac{1}{2},B=\frac{1}{2},C=0$$

• $$A=-\frac{1}{2},B=\frac{1}{2},C=\frac{1}{2}$$

• $$A=0,B=-\frac{1}{2},C=\frac{1}{2}$$

• $$A=\frac{1}{2},B=0,C=\frac{1}{2}$$

Successive Differences

Multiple Choice

Using the coefficients found by solving the system of equations, what is the closed form solution for the summation?

• $$\sum_{k=1}^nk=\frac{1}{2}n^2+\frac{1}{2}$$

• $$\sum_{k=1}^nk=\frac{1}{2}n^2+n$$

• $$\sum_{k=1}^nk=\frac{1}{2}n+\frac{1}{2}$$

• $$\sum_{k=1}^nk=\frac{1}{2}n^2-\frac{1}{2}n$$

The Closed Form Solution

Back to the Problem

Multiple Choice

Evaluate the limit:

$$\lim_{n\to\infty}\left(\frac{4}{n^2}\cdot\frac{n\cdot\ \left(n+1\right)}{2}\right)$$

• 1

• $$\infty$$

• 0

• 2

Back to the Problem

Stating the Solution

Don't Worry Too Much

This seems like a lot of work -- and that's because it actually is. We won't be working these problems by hand for very long. In the next few classes, we will start looking for shortcuts.

A Short Quiz!

Sorry, but I have to make sure you were paying attention!

Labelling

Label each part of an integral with the correct name.

Drag labels to their correct position on the image

Lower Limit of Integration

Integrand

Variable of Integration

Upper Limit of Integration

Integral Sign

Reorder

Reorder the steps to evaluate an integral correctly.

• Determine $$\Delta x$$

• Construct the partition

• Pull all constants out of the summation

• Find the closed form solution for the summation

• Evaluate the limit.

1

2

3

4

5

• 1
• 2
• 3
• 4
• 5

Categorize

Options (8)

$$f\left(x\right)=x^2$$  

$$f\left(x\right)=2^x$$  

$$f\left(x\right)=\left|x\right|$$  

$$f\left(x\right)=\sqrt{x+2}$$  

$$f\left(x\right)=\log x$$  

$$f\left(x\right)=\frac{1}{x}$$  

$$f\left(x\right)=-\sqrt{x}$$  

$$\text{sign}\left(x\right)$$  

Categorize each function based on whether or not it is continuous over the interval [-1,1]

Continuous

Discontinuous or Undefined

Multiple Choice

True or False: The left endpoint, midpoint, and right endpoint approximations are all examples of Riemann Sums.

• True
• False

• Only the midpoint approximation is an example of a Riemann Sum.

Practice Problems

Give these integrals a try! Feel free to ask any questions if you get stuck.

Hard Mode (Optional): Don't use a calculator.

Multiple Choice

Evaluate

$$\int_0^1\ x^2\ \text{d}x$$

• 1/4
• 1/3
• 1/2
• 2/3

Multiple Choice

Evaluate

$$\int_0^2\ x^2+2x\ \text{d}x$$

• 20/3

• 4

• 8/3

• 15/3

Multiple Choice

Evaluate

$$\int_{-3}^3\ x\ \text{d}x$$

• 6
• -3
• 0
• 3

Multiple Choice

Evaluate

$$\int_1^3\ x^3\ \text{d}x$$

• 20
• 30
• 25
• 15

Open Ended

What do you think may happen when we evaluate the following definite integral?

$$\int_0^x\ t\ \text{d}t$$

The End

This is the end of the lesson. There is a poll on the next three slides to provide feedback. Thanks guys!

Poll

"I think calculus is..."

• Like being in a boxing match against math itself.

• Actually worse than melted ice cream

• Maybe one of the neatest things I've ever seen?

• I want bacon.

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?