Definite Integrals
Definite Integrals
Section 5.2
OBJECTIVES: - be able to express the area under a curve as a definite integral and as a limit of Riemann sums - be able to compute the area under a curve using a numerical integration procedure - be able to make a connection with the definition of integration with the limit of a Riemann Sum
Sigma notation enables us to express a large sum in compact form:
[pic]
The Greek capital letter [pic](sigma) stands for “sum.” The index k tells us where to begin the sum (at the number below the [pic]) and where to end (the number above). If the symbol [pic] appears above the [pic], it indicates that the terms go on indefinitely. [pic] is called the norm of the partition which is the biggest [pic] (interval)
Riemann Sum: A sum of the form [pic] where f is a continuous function on a closed interval [a, b]; [pic] is some point in, and [pic] the length of, the kth subinterval in some partition of [a, b].
Big Ideas of a Riemann Sum:
- the limit of a Riemann sum equals the definite integral
- rectangles approximate the region between the x-axis and graph of the function
- A function and an interval are given, the interval is partitioned, and the height of each rectangle can be a value at any point in the subinterval
Negative area? Because the function is not positive, a Riemann sum does not represent a sum of areas of the rectangles.
It represents the sum of areas above the x-axis subtract the sum of areas below the x-axis. Area Under a Curve (as a Definite Integral)
If [pic] is nonnegative and integrable over a closed interval [a, b], then the area under the curve y = f(x) from a to b is the integral of f from a to b. [pic]
EX1: Evaluate each integral
EX1a: [pic] EX1b: [pic]
EX2: Use the graph of the integrand and areas to evaluate each integral.
EX2a: [pic] EX2b: [pic]
Section 5.2
OBJECTIVES: - be able to express the area under a curve as a definite integral and as a limit of Riemann sums - be able to compute the area under a curve using a numerical integration procedure - be able to make a connection with the definition of integration with the limit of a Riemann Sum
Sigma notation enables us to express a large sum in compact form:
[pic]
The Greek capital letter [pic](sigma) stands for “sum.” The index k tells us where to begin the sum (at the number below the [pic]) and where to end (the number above). If the symbol [pic] appears above the [pic], it indicates that the terms go on indefinitely. [pic] is called the norm of the partition which is the biggest [pic] (interval)
Riemann Sum: A sum of the form [pic] where f is a continuous function on a closed interval [a, b]; [pic] is some point in, and [pic] the length of, the kth subinterval in some partition of [a, b].
Big Ideas of a Riemann Sum:
- the limit of a Riemann sum equals the definite integral
- rectangles approximate the region between the x-axis and graph of the function
- A function and an interval are given, the interval is partitioned, and the height of each rectangle can be a value at any point in the subinterval
Negative area? Because the function is not positive, a Riemann sum does not represent a sum of areas of the rectangles.
It represents the sum of areas above the x-axis subtract the sum of areas below the x-axis. Area Under a Curve (as a Definite Integral)
If [pic] is nonnegative and integrable over a closed interval [a, b], then the area under the curve y = f(x) from a to b is the integral of f from a to b. [pic]
EX1: Evaluate each integral
EX1a: [pic] EX1b: [pic]
EX2: Use the graph of the integrand and areas to evaluate each integral.
EX2a: [pic] EX2b: [pic]