MAT.126 4.3

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MAT.126 4.3 Riemann Sums and Definite Integrals

Evaluate a definite integral using limits

Evaluating Definite Integrals

Apply the limit definition.

Use geometric formulas if applicable.

Use other techniques (coming soon).

Theorem 4.5 The Definite Integral as the Area of a Region

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 by

Area = S_a^b f(x) dx.

Theorem 4.4 Continuity Implies Integrability

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.

Definite vs. Indefinite Integrals

Indefinite integrals are families of functions.

Definite integrals are numbers.

Definition

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 by

lim_{||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.

Evaluate a definite integral using properties of definite integrals

Theorem 4.8 Preservation of Inequality

If f is integrable and nonnegative on the closed interval [a,b], then

0 <= 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.

Theorem 4.7 Properties of Definite Integrals

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], and

S_a^b kf(x) dx = k S_a^b f(x) dx

S_a^b [f(x) +- g(x)] dx

= S_a^b f(x) dx +- S_a^b g(x) dx

Theorem 4.6 Additive Interval Property

If f is integrable on the three closed intervals determined by a, b, and c, then

S_a^b f(x) dx

= S_a^c f(x) dx + S_c^b f(x) dx.

Definitions of Two Special Definite Integrals

Understand the definition of a Riemann sum

Norm

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.

Riemann Sum

Let f be defined on the closed interval [a,b], and let delta be a partition of [a,b] given by

a = x_0 < x_1 < x_2 < . . . < x_{n-1} < x_n = b

where 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 sum

E_{i=1}^n f(c_i) delta x_i, x_{i-1} <= c_i <= x_i

is called a Riemann sum of f for the partition delta.

Intervals of unequal width

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

Limits of sums

Area

Arc lengths

Average values

Centroids

Volumes

Work

Surface areas