The profile of a gear tooth shown in Fig. P9.1 (a) is approximated by the
(a) Estimate the area A using six rectangles of equal width
(b) Calculate the exact area by evaluating the definite
(a)
The estimated area of given function
Answer to Problem 1P
The estimated area of given function by rectangle method is
Explanation of Solution
Given:
The expression of profile of gear tooth is given as:
Number of rectangles given are 6.
Width of the rectangle is given as:
Concept used:
The expression of area is given as:
Here,
Calculation:
For
Substitute
Therefore,
For
Substitute
Therefore,
For
Substitute
Therefore,
For
Substitute
Therefore,
For
Substitute
Therefore,
Substitute
Therefore,
Substitute
Equation (3) can be written as:
Substitute
Conclusion:
Thus, the estimated area of the given function by rectangle method is
(b)
The exact area of given function
Answer to Problem 1P
The exact area of given function by integration method is
Explanation of Solution
Concept used:
The expression of area is given as:
Here,
Calculation:
Substitute
Further simplify for
Conclusion:
Thus, the exact area of given function by integration method is
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