5.3 The Definite Integral and the Fundamental Theorem of Calculus

“When faced with something you don’t know how to handle, try to relate it to something you do know how to handle”

The Area of a region under a curve - can be viewed as sum of many rectangles –

Pictures on pages 399 and 400

The Definite Integral f(x) a continuous function on [a,b]. Subdivide the interval into n equal parts, each of width . A number x_k from the kth subinterval (for k = 1, 2, 3, ... n)

Riemann Sum

The Definite Integral

Terminology:

Integrand – f(x) Upper and lower limits – a and b

Area as Definite Integral - If f(x) is continuous and on the interval , then the region R under the curve over the interval has area A given by the definite integral

Fundamental Theorem of Calculus

If the function f(x) is continuous on [a,b], then

where F(x) is any antiderivative of f(x) on [a,b]

Examples:

RULES – Similar to indefinite integral for constant multiple, sum and difference

Additional Rules

Examples:

Example

If find

#46 Find the area of the region R that lies under the given curve y = f(x) over the indicated interval [a,b]

Under

Applications

MARGINAL COST

The marginal cost of producing a certain commodity is C’(q) = 6q +1 dollars per unit when q units are being produced.

a) What is the total cost of producing the first 10 units.

b) What is the cost of producing the next 10 units?

#52 WATER POLLUTION

It is estimated that t years from now the population of a certain lake-side community will be changing at the rate of thousand people per year. Environmentalists have found that the level of pollution in the lake increases at the rate of approximately 5 units per 1,000 people. By how much will the pollution in the lake increase during the next 2 years?