- \n
- Differentiation<\/strong> \u2014 as we saw last term, differentiation allows us to compute and study the instantaneous rate of change of quantities. At its most basic it allows us to compute tangent lines<\/a> and velocities, but it also led us to quite sophisticated applications including approximation of functions through Taylor polynomials<\/a> and optimisation of quantities by studying critical and singular points.<\/li>\n
- Integration<\/strong> \u2014 at its most basic, allows us to analyse the area under a curve. Of course, its application and importance extend far beyond areas and it plays a central role in solving differential equations<\/a>.<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/www.booksofall.com\/cn\/wp-content\/uploads\/sites\/2\/2023\/02\/img_63fd6108a6ac5.png\")
<\/p>\nIt is not immediately obvious that these two topics are related to each other. How- ever, as we shall see, they are indeed intimately linked.<\/p>\n
1.1 Definition of the Integral<\/h3>\n
Arguably the easiest way to introduce integration is by considering the area between the graph of a given function and the x-axis, between two specific vertical lines \u2014 such as is shown in the figure above. We\u2019ll follow this route by starting with a motivating example.<\/p>\n
1.1.1\u00a0 A Motivating Example<\/strong><\/p>\n
Let us find the area under the curve y = ex (and above the x-axis) for 0 \u2264 x \u2264 1. That is, the area of { (x, y) \u2223 0 \u2264 y \u2264 ex, 0 \u2264 x \u2264 1 } .<\/p>\n
![\"\"](\"https:\/\/www.booksofall.com\/cn\/wp-content\/uploads\/sites\/2\/2023\/02\/img_63fd616d7b90d.png\")
<\/p>\n - Integration<\/strong> \u2014 at its most basic, allows us to analyse the area under a curve. Of course, its application and importance extend far beyond areas and it plays a central role in solving differential equations<\/a>.<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/www.booksofall.com\/cn\/wp-content\/uploads\/sites\/2\/2023\/02\/img_63fd617a000da.png\")
<\/p>\n