Integration

You will learn the definition of integration by Riemann sums. Then we will discuss the fundamental theorem of calculus to define the relation between integral and derivative.

Context: energy consumption

The pret-a-loger team changed an existing house into a sustainable one to reduce energy consumption. But how can you measure energy consumption? Watch the video and see what integration has to do with it!

The concept of integration

To measure the energy consumption you can look at the power that is used times a time-interval. By taking smaller and smaller intervals you can get the Riemann sum. This process, of taking the limit of sums for smaller and smaller intervals, is called integration.

The meaning of the integral

What is the meaning of the integral? Is it the area underneath the graph? If you think so, please watch this video and learn about the signed area!

Fundamental theorem of Calculus

How can you evaluate integrals exactly? The extremely powerful tool for this is called: the fundamental theorem of Calculus. This video reviews the exact statement of the theorem and shows how integration and differentiation are related.

Proof of fundamental theorem of Calculus

The fundamental theorem of Calculus says roughly that integration and differentiation are inverse operations. This video shows you why the fundamental theorem of Calculus is true.

Rules of calculation for integration - part 1

Speed is the derivative of distance travelled. Using the Fundamental Theorem of Calculus it follows that distance travelled is the integral of speed as a function of  time. Using this example you will understand the rules of calculation for integration, when the integrand is multiplied by a constant or two integrands are added.

Rules of calculation for integration - part 2

Some useful properties of the intervals of integrals are shown in this video. The integral of the piece-wise defined function |x| (absolute value of x) is used as an example.

Rules of calculation for integration - part 3

We learned that there is no general rule for calculating integrals of compositions of  functions. In this video we show how to calculate integrals with a linear function within a more complex function using linear substitution.

Integration by substitution

Learn the method of substitution for calculating anti-derivatives for functions that are combinations of elementary functions.

The substitution method

There are several integration techniques to help evaluate a given integral. One of these techniques is the method of substitution. This method will be discussed in this video. 

Substitution with boundaries

After learning how to use substitution in integrals without boundaries this video will show how to use substitution in integration with boundaries.

Integration by parts

In this video you will learn another major technique for calculating integrals: integration by parts. This will allow you to deal with integrals which are products of two functions.

Integration by parts - part 1

Calculating the integral of the product of two functions can become a lot easier with integration by parts. In this video integration by parts is derived from the product rule and applied an example.

Integration by parts - part 2

To apply integration by parts, you need to make several choices. In this video, some tips and tricks are given that can help you when applying integration by parts.

Summary about integration

What is the definition of integration ? How are integration and differentiation linked through the fundamental theorem of Calculus? What are the rules of calculation for evaluating integrals? This video gives the summary of week 6 of the edX course: pre-university Calculus (Calc001x).

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