Integrals

One of the fundamental problems in calculus is the computation of the area between the graph of a function f : [ a ,   b ] → ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152721/7eeee3ed-efa4-434d-8da2-13dce29a52aa/content/eq1027.tif"/> and the x-axis. The essential ideas are illustrated in Figure 6.1 and Figure 6.2. An interval [a, b] is divided into n subintervals [x_k, x _k+1], with a = x ₀ < x ₁ < · · · < x_n = b. On each subinterval the area is approximated by the area of a rectangle, whose height is usually the value of the function f(t_k ) at some point t_k ∈ [x_k, x _k+1]. In the left figure, the heights of the rectangles are given by the values f(x_k ), while on the right the heights are f(x _k+1). FIGURE 6.1 Graph of 1 + <italic>x</italic> + sin(<italic>x</italic>) with lower Riemann sums. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152721/7eeee3ed-efa4-434d-8da2-13dce29a52aa/content/fig6_1.tif"/> FIGURE 6.2 Graph of 1 + <italic>x</italic> + sin(<italic>x</italic>) with upper Riemann sums. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152721/7eeee3ed-efa4-434d-8da2-13dce29a52aa/content/fig6_2.tif"/>