The definite integral of a function f on the interval from a to b, where a < b, is a unique number I that is greater than or equal to all the lower sums and less than or equal to all of the upper sums of that function. The lower and upper sums of a function f for a partition P of [a, b] are defined as L_f(P) = m₁x₁ + m₂x₂ + ... m_nx_n and U_f(P) = M₁x₁ + M₂x₂ + ... M_nx_n, where for any integer within a subinterval the ms and Ms are the minimum and maximum values of f on that particular subinterval. A definite integral is usually written as ^b_af(x)dx. The is called the integral sign, the numbers a and b are referred to as the limits of integration, and...