University Mathematics Handbook (2015)
V. Integral Calculus of Single-Variable Functions
Chapter 2. Definite Integral
2.1 Definition
Let be a function defined on interval . Let's divide interval into subintervals by the points
We denote , using the same denotation for the length of all subintervals, , .
Let us denote . In every , we select an arbitrary point . The expression
is called Riemann sum, according to the partition to subintervals and the selected points .
Definition: function is called a Riemann integrable function, on interval , if the limit exists and is not dependent on the selection of partitions and points . We denote this limit by , and it is called the definite integral of function on interval , when and in integral are called limits of integration.
2.2 Classes of Integrable Functions
a. If function is not bounded on interval , then it is not integrable on that interval.
b. If function is continuous in interval , then it is integrable in it.
c. A function defined and bounded in interval is called piecewise continuous function, if it has at most a finite set of discontinuities, all of which are jump discontinuities.
d. If function is piecewise continuous in interval , then it is integrable in it.
2.3 Properties of Definite Integral
a.
b.
c. for every constant
d.
e. If , then , particularly when , .
f. If and are integrable on , then is integrable on interval .
g. Mean Value Theorem for Integrals
If is continuous on , then there is a point on , such that
The number is called the average value of on .
h. If for all , then .
i.
1. If is integrable on , then is also integrable in that interval and there holds .
2. The inverse is incorrect. That is, if is integrable, it doesn't necessarily follow that is integrable in that interval.
j. If and for all , then
2.4 Connection Between the Indefinite and Definite Integral
If is a continuous function and is its antiderivative on , then:
a. , when
b.
c. Newton-Leibniz Formula
d.
e. Cauchy-Schwartz Inequality
If and are integrable on , then:
2.5 Calculating Definite Integrals
a. Change of Variables
If function is continuous on , and if function is continuously differentiable on , if its image equals interval , and there holds and , then:
b. Integration by Parts
If functions and have continuous derivatives on , then:
c. Integral of Even and Odd Functions on Interval
If function is even and integrable on , then:
And, if function is odd and integrable on , then:
2.6 Numerical Methods of Computing Definite Integrals
We divide interval into equal sub-intervals, by points:
when , .
a. The following formulas are rectangle approximations:
when is the midpoint of .
The approximation error, in the case that is twice differentiable is .
b. Trapezoid Approximation
The approximation error is
c. Simpson's Rule
when is then midpoint of .
The approximation error is
2.7 Applications of Definite Integral
a. The area of a plane confined by the graph of non-negative function , -axis, and straight lines and is
b. The Area of a plane bounded by a curve presented in the parametric form is
c. The area of a plane bounded by two functions , , above interval is
d. The area of a plane bounded by a curve presented by , , and two rays, and is
e. The length of a planar curve given by:
1) , is
2) , is
f. The volume of a solid of revolution around the -axis of an area confined by the graph of non-negative function and the straight lines , , is
g. The volume of a solid of revolution around the -axis of an area confined by the graphs of functions and and straight lines , , is
h. The volume of a solid of revolution around the -axis of an area confined by the graph of non-negative function and the straight lines , , is
i. The volume of a solid of revolution around the -axis of an area confined by the graphs of functions and in interval is
j. If a solid body is confined by planes , , and the plane carves out of the body an area for every , then, the volume of the body is
Example: Calculate the volume of the sphere , confined between the planes and .
For every situated at , the section is circle , the area of which is . Therefore, the volume of the sphere is