Chapter 3: Calculus~Hughes-Hallett The Definite Integral

Area Approximation: Left-Hand Sum Width of rectangle: Δt Length of rectangle: f(t) Area = A, ΔA = f(t) Δt

Area Approximation: Right-Hand Sum , Approximate Total Area = 406.25 Width of rectangle: Δt Length of rectangle: f(t) Area = A, ΔA = f(t) Δt

Approximate Error E(x)=[f(b)-f(a)]t  t = 1, a = 0, b = 10  t = .5, E(x) = 15  t = .25, E(x) = 7.5 Approximate Error

Exact Area Under the Curve The Definite Integral gives the exact area under a continuous curve y = f(x) between values of x on the interval [a,b].

The Definite Integral Physically - is a summing up Geometrically - is an area under a curve Algebraically - is the limit of the sum of the rectangles as the number increases to infinity and the widths decrease to zero:

The Definite Integral as an AREA When f(x) > 0 and a < b: the area under the graph of f(x), above the x-axis and betweeen a and b = When f(x) > 0 for some x and negative for others and a < b: is the sum of the areas above the x-axis, counted positively, and the areas below the x-axis , counted negatively.

The Definite Integral as an ALGEBRAIC SUM When f(x) > 0 for some x and negative for others and a < b: is the algebraic sum of the positive and “negative” areas formed by the rectangles and is, therefore, not the total area under the curve!

Notation for the Definite Integral Since the terms being added up are products of the form: f(x) •x the units of measure- ment for is the product of the units for f(x) and the units for x; e.g. if f(t) is velocity measured in meters/sec and t is time measured in seconds, then has units of (meters/sec) • (sec) = meters.

The Definite Integral as an AVERGE The average value of a function f(x) from a to be is defined as:

The Fundamental Theorem of Calculus (Part 1) If f is continuous on the interval [a,b] and f(t) = F’(t), then: In words: the definite integral of a rate of change gives the total change.

Concrete Example of the FTC: The area under the curve f(x) = 2x from xo to x1 is the y value of F(x1) - F(x0), while the slope of F(x) = x2 at x = x1 is F’(x1) = f(x1) F(x) = x2, F’(x) =f(x) = 2x The equation of the tangent line to y = x2 at (4,16) is y = 8x - 16 and the slope of the tangent line is 8. f(x) = F’(x) = 2x, F(x) = x2 The area under f(x) from x = (0,0) to (4,8) is the value of F(x) = x2 at x = 4, i.e. F(4) - F(0) = 16.

Theorem: Properties of Limits of Integration If a, b, and c are any numbers and f is a con- tinuous function, then: 1. 2. In words: 1. The integral from b to a is the negative of the integral from a to b. 2. The integral from a to c plus the integral from c to b is the integral from a to b.

Theorem: Properties of Sums and Constant Multipliers of the Integrand Let f and g be continuous functions and let a, b and c be constants: 1. 2. In Words: 1. The integral of the sum (or difference) of two func- tions is the sum (or difference ) of their integrals. 2. The integral of a constant times a function is that constant times the integral of the function.

Theorem: Comparison of Definite Integrals Let f and g be continuous functions and suppose there are constants m and M so that: We then say f is bounded above by M and bound- ed below by m and we have the following facts: 1. 2.

Stay Tuned! More follows. Are you curious?

Mathematical Definition of the Definite Integral Suppose that f is bounded above and below on [a,b]. A lower sum for f in the interval [a,b] is a sum: where is the greatest lower bound for f on the i-th interval. An upper sum is: where is the least upper bound for f on the i-th interval. The definition of the Definite Integral: Suppose that f is bounded above and below on [a.b]. Let L be the least upper bound for all the lower sums for f on [a,b], and let U be the greatest lower bound for all the upper sums. If L = U, then we say that f is integrable and we define to be equal to the common value of L and U!

Two Theorems on Integrals: The Mean Value Equality for Integrals: Continuous Functions are Integrable: If f is continuous on [a,b], then exists.

The General Riemann Sum A general Riemann sum for f on the interval [a,b] is a sum of the form: where