Mathematical Investigation Mathematics & Economics Math Problem (Math Problem Sample)

Integration Techniques and Applications
Name
Institution
Integration Techniques and Applications
Introduction
The objective of this investigation wasto model the transverse cross sections of a champagne flute and wine glass using trigonometric function QUOTE

 for the region QUOTE

. Also, the investigation was used todetermine how the shape of an object could be altered by changing the variables of the trigonometric function. During the investigation, the bowl of the champagne was modelled such that its cross section area was approximately equal to the cross sectional area of the wine glass.
From the investigation, it was found that the volume of the function  QUOTE

 could not be accurately determined by adding the volume of the cone to that of the truncated cone because some parts of the function were not included when calculating the volume. It was also found that the function QUOTE

, used to model the shape of the champagne glass, could not produce a shape that was reasonable enough to be used as a wine glass.
Task 1
The wine glass bowl was modelled using the trigonometric function shown below.


Since the function passes through the points QUOTE

, the values of a an
d b in the function were calculated by substituting the points on the model equation.
Substituting the  QUOTE




























The function  QUOTE

 produced the graph shown in the figure 1 below.


The function  QUOTE

 is symmetric about  QUOTE HYPER14HYPER15 because
Proving symmetry about:
Where
The function is symmetrical if.
The function is symmetry about π because.
Task 2
To approximate the area of the cross section of the bowl, we find the sum of the area of the trapezium and that of the isosceles triangle shown on the figure 2 below.


Figure  SEQ Figure \* ARABIC 1
Area of the trapezium