In this video I dive deeper in the wonderful world of integration and solve some very useful examples on using the fundamental theorem of calculus which I showed in my earlier videos. I also provide a brief history lesson on how it was very difficult to solve areas prior to the invention of this theorem. This is an important video on integrals so make sure you watch it.

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Fundamental Theorem of Calculus: Examples

Example 1

Example 2

Example 3

Example 4

Notes on Example 4

• The French mathematician Gilles de Roberval first found the area under the sine and cosine curves in 1635, this was a very challenging problem that required a great deal of ingenuity.
• If we didn’t have the benefit of the Fundamental Theorem, we would have to compute a difficult limit of sums using obscure trigonometric identities (or a computer algebra system to solve the limit)
• It was even more difficult for Roberval because the apparatus of limits had not been invented in 1635.
• But in the 1660s and the 1670s when the Fundamental Theorem was discovered by Isaac Barrow and exploited by Newton and Leibniz, such problems became very easy, as you can see from Example 4.