Equation of curve: \(y = f(x)\) for \(a \leq x \leq b\)

We assuem that \(f\) is continous on \([a,b]\) and \(f(x) \geq 0\) for all \(x \in [a,b]\)

Let \(A\) denote its area. To approximate \(A\), we choose a positive integer \(n\) and cut the interval \([a,b]\) into \(n\) pieces, each of width \(\Delta x = \dfrac{b-a}{n}\). The dividing points between these pieces are the numbers \(x_i = a + i\Delta x\) for \(i=0,1,2,...,n\) and for \(1 \leq i \leq n\) the ith piece is the interval \([x_{i-1}, x_i]\).

If we let \(A_i\) denote the area of the ith strip, which is the region defined by the inequalities \(x_i-1 \leq x \leq x_i\) and \(0 \leq y \leq f(x)\), then \(A = A_1 + A_2 + ... A_n = \sum_{i=1}^n A_i\)

\(R_n = \sum_{i=1}^n f(x_i^{*})\Delta x\)

Area = \(\lim_{n \to \infty} R_n = \int_a^b f(x) \mathrm{d}x\)

Suppose that \(f\) and \(g\) are both continous on the interval \([a,b]\) and for all \(x \in [a,b]\), \(g(x) \leq f(x)\).

We want to compute the area \(A\) of the region \(R\) between the graphs of \(f\) and \(g\) for \(x \in [a,b]\), that is the area of the region defined by the inequalities \(a \leq x \leq b\) and \(g(x) \leq y \leq f(x)\)

\(R_n = \sum_{i=1}^n (f(x_i^{*}) - g(x_i^{*}))\Delta x\)

\(A = \lim_{n \to \infty} R_n = \int_a^b (f(x) - g(x))\mathrm{d}x\)