Suppose that \(a < b\), \(f\) is a function that is defined on the closed interval \([a,b]\), and \(n\) is a positive integer. Let \(\Delta x = \dfrac{b-a}{n}\), and for \(0 \leq i \leq n\) let \(x_i = a + i\Delta x\). These numbers divide the interval \([a,b]\) into \(n\) smaller intervals \([x_0, x_1], [x_1, x_1], ..., [x_{n-1}, x_n]\), each of width \(\Delta x\). Also, suppose that for \(1 \leq i \leq n, x^{*}_i \in [x_{i-1}, x_i]\). Then the sum

\(f(x^{*}_1)\Delta x + f(x^{*}_2)\Delta x + ... + f(x^{*}_n)\Delta x = \sum_{i=1}^nf(x^{*}_i)\Delta x\)

is called a Riemann sum for \(f\) on the interval \([a,b]\). We may also call it a Riemann n-sum if we want to specify the number of terms in the sum.

Suppose that \(a < b\) and \(f\) is a function that is continous on the closed interval \([a,b]\). Then every Riemann sum sequence for \(f\) on \([a,b]\) converges, and all such sequece converge to the same number. In other words, there is some number \(L\) such that if \((R_n)^{\infty}_{n=1}\) is any Riemann sum sequence for \(f\) on \([a,b]\), then \(\lim_{n \to \infty} R_n = L\).

Suppose that \(a < b\) and \(f\) is continous on \([a,b]\). Then the number \(L\) that is the limit of all Riemann sum sequences for \(f\) on \([a,b]\) is called the definite integral of \(f\) from \(a\) to \(b\), and it is denoted \(\int_a^b f(x) \mathrm{d} x\). In other words, for any Riemann sum sequence \((R_n)^{\infty}_{n=1}\),

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

• The number \(a\) is called the lower limit of integration (Not to be confused with the standard meaning of limit)
• The number \(b\) is called the upper limit of integration
• \(f(x)\) is called the integrand

More informally, in the notation of Definition 5.3.1 we can say that for large \(n\),

\(\int_a^b f(x) \mathrm{d} x \approx \sum_{i=1}^n f(x_i^{*})\Delta x\)

• The symbol \(\int\) is called an integral sign. It is an elongated \(S\) which is intended to remind you that the expression on the right is a sum.
• The expression \(f(x) \mathrm{d}x\) on the left is supposed to remind you of the formula \(f(x_i^{*})\Delta x\) that appears in the Riemann sum on the right.
• The \(\mathrm{d}x\) at the end of the integral notation is to identify the independent variable. (They also play an imporant role in the method of integration by substitution)
• Geometrical interpretation:
  • If \(f(x)\) is positive for all values of \(x\), then \(\int_a^b f(x) \mathrm{d}x\) is equal to the area of the region under the graph of \(f\) for \(a \leq x \leq b\)
  • If \(f(x)\) is not always positive, then we divide the region between the graph of \(f\) and the \(x\) axis for \(a \leq x \leq b\) into two parts, the part above the x-axis and the part below. The value of \(\int_a^b f(x) \mathrm{d}x\) is equal to the area of the part above the x-axis minus the area of the part below.

Suppose that \(a < b\), \(f\) and \(g\) are continous on \([a,b]\) and \(c\) is any real number. Then:

• \(\int_a^b cf(x) \mathrm{d}x = c\int_a^b f(x) \mathrm{d}x\)
• \(\int_a^b (f(x) + g(x)) \mathrm{d}x = \int_a^b f(x) \mathrm{d}x + \int_a^b g(x) \mathrm{d}x\)
• \(\int_a^b (f(x) - g(x)) \mathrm{d}x = \int_a^b f(x) \mathrm{d}x - \int_a^b g(x) \mathrm{d}x\)

Suppose that \(a < b\) and \(f\) is continous on \([a,b]\). Let \(m\) be the minimum value of \(f\) on \([a,b]\), and let \(M\) be the maximum value. Then

\(m(b-a) \leq \int_a^b f(x) \mathrm{d}x \leq M(b-a)\)

Suppose that \(a < b\), \(f\) and \(g\) are continous on \([a,b]\), and for all \(x \in [a,b], f(x) \leq g(x)\). Then \(\int_a^b f(x) \mathrm{d}x \leq \int_a^b g(x) \mathrm{d}x\)

Suppose that \(a < b < c\), and \(f\) is continous on \([a,c]\). Then

\(\int_a^b f(x) \mathrm{d}x + \int_b^c f(x) \mathrm{d}x = \int_a^c f(x) \mathrm{d}x\)

For \(a \geq b\), we defined \(\int_a^b f(x) \mathrm{d}x\) as follows:

• If \(a = b\) then \(\int_a^b f(x) \mathrm{d}x = 0\)
• If \(a > b\) then \(\int_a^b f(x) \mathrm{d}x = - \int_b^a f(x) \mathrm{d}x\)

Suppose \(f\) is continous on an interval \(I\), and \(a,b\), and \(c\) are numbers in I. Then

\(\int_a^b f(x) \mathrm{d}x + \int_b^c f(x) \mathrm{d}x = \int_a^c f(x) \mathrm{d}x\)

Suppose that \(f\) and \(g\) are continous on an interval \(I\), \(a\) and \(b\) are numbers in \(I\), and \(c\) is any number. Then:

• \(\int_a^b cf(x) \mathrm{d}x = c\int_a^b f(x) \mathrm{d}x\)
• \(\int_a^b (f(x) + g(x)) \mathrm{d}x = \int_a^b f(x) \mathrm{d}x + \int_a^b g(x) \mathrm{d}x\)
• \(\int_a^b (f(x) - g(x)) \mathrm{d}x = \int_a^b f(x) \mathrm{d}x - \int_a^b g(x) \mathrm{d}x\)