Chapter 5: Calculus~ Hughes-Hallett

1. Chapter 5: Calculus~Hughes-Hallett • The Definite Integral

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

3. 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

4. ApproximateError • E(x)=[f(b)-f(a)]t •  t = 1, a = 0, b = 10 • E(x) = (50 - 20)*1 • E(x) = 30 •  t = .5, E(x) = 15 •  t = .25, E(x) = 7.5

5. 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].

6. 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:

7. 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.

8. 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!

9. 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.

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

11. 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.

12. 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. 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. 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)

13. 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.

14. 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.

15. 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.

16. Stay Tuned! • More follows. • Are you curious?

17. 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!

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

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