## ABSTRACT

We wish to study the concepts of length, area, volume, as well as higher dimensional analogues and abstractions of these ideas quite systematically. In 1, 2 or 3 dimensions, we already know how to use the Riemann integral to calculate the area or the volume 1 of certain regions of ℝ2 and ℝ3. For example, we know how to calculate the area under the graph of “nice” (differentiable or at least continuous) functions (see Figure 6.1). The grey area <italic>A</italic> between the graphs of the “nice” functions <italic>f, g</italic> can be calculated by the integral <inline-formula> <alternatives> <mml:math display="inline" xmlns:mml="<a href="https://www.w3.org/1998/Math/MathML" target="_blank">https://www.w3.org/1998/Math/MathML</a>"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mstyle displaystyle="true"> <mml:mrow> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mn>0</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:mrow> </mml:mstyle> <mml:mtext>d</mml:mtext> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> <inline-graphic xlink:href="<a href="https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315195865/b46e112d-cbc8-4dce-b9d6-367bdebe9455/content/eq1596.tif" target="_blank">https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315195865/b46e112d-cbc8-4dce-b9d6-367bdebe9455/content/eq1596.tif</a>" xmlns:xlink="<a href="https://www.w3.org/1999/xlink" target="_blank">https://www.w3.org/1999/xlink</a>"/> </alternatives> </inline-formula>. https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315195865/b46e112d-cbc8-4dce-b9d6-367bdebe9455/content/fig6_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>