# Introduction to Integration

### Integration as Summation

### Video

This *Khan Academy* video introduces the integral as the limit of Riemann sums used to approximate the area under a curve.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

### Notes

These notes from MathCentre give a geometric description of integration as the limit of a summation used to compute the area under a curve.

A pdf of this document can be downloaded here.

### Interactive Exercises

This NUMBAS exercise asks you to compute Riemann sums of a quadratic function.

Please click here to view this NUMBAS question in another window.

### Rules of Integration

### Video

This video from *Firefly Lectures* gives a comprehensive list of the fundamental rules of integration.

Note: All *Firefly Lectures* content is available for free at www.fireflylectures.com.

### Notes

These notes from MathCentre go through the most common rules you will need when working on integration problems.

A pdf of this document can be downloaded here.

### Interactive Exercises

This NUMBAS exercise lets you practise what you’ve learned about the rules of integration.

Please click here to view this NUMBAS question in another window.

### Definite Integrals

### Videos

This video from *Khan Academy* discusses the idea and notation behind the definite integral.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

This video from *Firefly Lectures* gives an introductory example of computing a definite integral.

Note: All *Firefly Lectures* content is available for free at www.fireflylectures.com.

This video from *Khan Academy* shows how to compute the area under the curve using definite integrals.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

### Notes

These notes from MathCentre discuss the notation for indefinite integrals and give some examples of how they are computed.

A pdf of this document can be downloaded here.

### Interactive Exercises

### The Fundamental Theorem of Calculus

### Videos

This video from *Khan Academy* explains how the first part of the Fundamental Theorem of Calculus connects integration and differentiation, then shows how it can be used to compute the derivative of complex definite integrals.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

This video from *Firefly Lectures* shows how to use the Fundamental Theorem of Calculus to compute derivatives of complex integral expressions. Note: What is referred to in this video as the Second Fundamental Theorem of Calculus is generally known in Ireland as the First (Part of the) Fundamental Theorem of Calculus.

Note: All *Firefly Lectures* content is available for free at www.fireflylectures.com.

This video from *MIT OpenCourseWare* shows how to compute integrals where the upper limit is a function of $x$ using the Fundamental Theorem of Calculus.

Instructor: Christine Breiner. Used under a Creative Commons BY-NC-SA Licence. More information is available at http://ocw.mit.edu/terms.

This video from *patrickJMT* discusses the second part of Fundamental Theorem of Calculus and how it is used to compute definite integrals.

Note: All *patrickJMT* content is available for free at www.youtube.com/user/patrickJMT.

Patrick JMT’s Website: www.patrickjmt.com.

### Notes

These notes from MathCentre discuss the Fundamental Theorem of Calculus and how it relates differentiation and integration. The two parts of the statement of the theorem are given in the key points of the notes.

A pdf of this document can be downloaded here.