what is function in math's problems on math's

1. What are the Functions?
Functions play a vital role in Mathematics. It is
defined as a special association between the set of
input and output values in which each input value
correlates one single output value. We know that
there are two types of Euler integral functions. One
is a beta function, and another one is a gamma
function. The domain, range or co-domain of
functions depends on its type. In this page, we are
going to discuss the definition, formulas, properties,
and examples of beta functions.

2. The beta function formula is defined as follows:
Where p, q > 0
The gamma function is defined as:
Relation with Gamma Function

3. Beta Function Applications
In Physics and string approach, the beta
function is used to compute and
represent the scattering amplitude for
Regge trajectories. Apart from these,
you will find many applications in
calculus using its related gamma
function also.

4. The definite integral is the key tool in
calculus for defining and calculating areas
and volumes.
We also use it to compute quantities such as
the lengths of curved paths, probabilities,
averages, energy consumption, the mass of
an object, and the force against a dam’s
floodgates, to name only a few.

11. Classification of Integral Formulas
Rational functions
Irrational functions
Trigonometric functions
Inverse trigonometric functions
Hyperbolic functions
Inverse hyperbolic functions
Exponential functions
Logarithmic functions
Gaussian functions

13. A pyramid 3 m high has a square base that is 3 m on a side. The cross section
of the pyramid perpendicular to the altitude x m down from the vertex is a square
x m on a side. Find the volume of the pyramid.

14. 1. A sketch. We draw the pyramid with its altitude along the x-axis and its
vertex at the
origin and include a typical cross-section

15. A curved wedge is cut from a circular cylinder of radius 3 by two planes.
One plane is perpendicular to the axis of the cylinder. The second plane crosses
the first
plane at a 45° angle at the center of the cylinder. Find the volume of the wedge.

22. We have a cable that weighs 2 lbs/ft attached to a bucket filled with coal that weighs
800 lbs. The bucket is initially at the bottom of a 500 ft mine shaft. Answer each of
the following about this. (a) Determine the amount of work required to lift the bucket
to the midpoint of the shaft. (b)Determine the amount of work required to lift the
bucket from the midpoint of the shaft to the top of the shaft.

26. [6]

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29. The line segment x = 1- y where y= 0 to 1 is revolved about the y-axis to
generate the cone in Figure. Find its lateral surface area (which excludes the
base area).
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30. Your company decided to put out a deluxe version of a wok you designed. The plan is
to coat it inside with white enamel and outside with blue enamel. Each enamel will be
sprayed on 0.5mm thick before baking. (See accompanying figure.) Your
manufacturing department wants to know how much enamel to have on hand for a
production run of 5000 woks. What do you tell them? (Neglect waste and unused
material and give your answer in liters. Remember that 1 cm= 1mL so 1L = 1000 cm).
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