MAT.126
4.3
Riemann Sums and Definite Integrals

AreaArc lengthsAverage valuesCentroidsVolumesWorkSurface areas

When taking a limit, it's the width of the largest subinterval that must go to zero.

Let f be defined on the closed interval [a,b], and let delta be a partition of [a,b] given bya = x_0 < x_1 < x_2 < . . . < x_{n-1} < x_n = bwhere delta x_i is the width of the ith subinterval. If c_i is any point in the ith subinterval [x_{i-1},x_i], then the sumE_{i=1}^n f(c_i) delta x_i, x_{i-1} <= c_i <= x_iis called a Riemann sum of f for the partition delta.

The width of the largest subinterval of a partition delta is the norm of the partition and is denoted by ||delta||.A partition in which all of the subintervals are of equal width is regular.

If f is defined at x=a, then we define S_a^a f(x) dx = 0.If f is integrable on [a,b], then we define S_b^a f(x) dx = -S_a^b f(x) dx.

If f is integrable on the three closed intervals determined by a, b, and c, thenS_a^b f(x) dx = S_a^c f(x) dx + S_c^b f(x) dx.

If f and g are integrable on [a,b] and k is a constant, the functions kf and f +- g are integrable on [a,b], andS_a^b kf(x) dx = k S_a^b f(x) dxS_a^b [f(x) +- g(x)] dx = S_a^b f(x) dx +- S_a^b g(x) dx

If f is integrable and nonnegative on the closed interval [a,b], then0 <= S_a^b f(x) dx.If f and g are integrable on the closed interval [a,b] and f(x) <= g(x) for every x in [a,b], then S_a^b f(x) dx <= S_a^b g(x) dx.

If f is defined on the closed interval [a,b] and the limit of Riemann sums over partitions delta exists, then f is said to be integrable on [a,b] and the limit is denoted bylim_{||delta||-->0} E_{i=1}^n f(c_i) delta x_i= S_a^b f(x) dx.The limit is called the definite integral of f from a to b. The number a is the lower limit of integration, and the number b is the upper limit of integration.

Indefinite integrals are families of functions.Definite integrals are numbers.

If a function f is continuous on the closed interval [a,b], the f is integrable on [a,b]. That is, S_a^b f(x) dx exists.

If f is continuous and nonnegative on the closed interval [a,b], then the area of the region bounded by the graph of f, the x-axis, and the vertical lines x=a and x=b is given byArea = S_a^b f(x) dx.

Apply the limit definition.Use geometric formulas if applicable.Use other techniques (coming soon).