RIEMANN SUMS AND DEFINITE INTEGRALS Section 4.3

When you are done with your homework, you should be able to… Understand and use the definition of a Riemann sum Evaluate a definite integral using limits Evaluate a definite integral using properties of definite integrals

Kepler lived from 1570-1640 in Poland Kepler lived from 1570-1640 in Poland. He discovered 3 laws of planetary motion: Orbits are ellipses, the time around the earth squared is directly proportional to the average distance to the sun cubed, and equal areas are swept out in equal time. Why are the 3 laws significant? They confirmed the Pythagorean Philosophy of nature. They confirmed the value of empiricism. The led to the discovery of “dark matter”. All of the above.

RIEMANN SUMS Let f be defined on the closed interval and let be a partition of given by where is the width of the ith subinterval. If is any point in the ith subinterval, then the sum is called a Riemann Sum of f for the partition .

NORM OF THE PARTITION The width of the largest subinterval of a partition is the norm of the partition and is denoted by .

REGULAR PARTITION If every subinterval is of equal width, the partition is regular and the norm is denoted by

RELATIONSHIP BETWEEN THE NORM AND THE NUMBER OF SUBINTERVALS For a general partition, the norm is related to the number of subintervals of in the following way: So the number of subintervals in a partition approaches infinity as the norm of the partition approaches zero.

DEFINITE INTEGRALS Consider the following limit: To say that this limit exists means that for every there exists a such that for every partition with it follows that

Definition of a Definite Integral If f is defined on the closed interval and the limit exists, then f is integrable on and the limit is denoted by 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.

What is the difference between definite integrals and indefinite integrals? A definite integral is a family of functions and an indefinite integral is a number. A definite integral is a number and an indefinite integral is a family of functions. There is no difference between a definite integral and an indefinite integral.

Evaluate the definite integral

Theorem: The definite Integral as the Area of a Region If f is continuous and nonnegative on the closed interval , then the area of the region bounded by the graph of f, the x-axis, and the vertical lines and is given by

Good news, Bad news… The good news is that in the next section we will be learning how to directly evaluate definite integrals. The bad news is that for now you must use The limit definition, Or you can check to see whether the definite integral represents the area of a common geometric region such as a rectangle, triangle, or semicircle.

Definitions of Two Special Definite Integrals If is defined at , then we define If is integrable on , then we define

Theorem: Preservation of Inequality If f is integrable and nonnegative on the closed interval , then If f and g are integrable on the closed interval and for every x in , then  

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

Properties of Definite Integrals If f and g are integrable on and k is a constant, then the functions of and are integrable on , and and