Concept explainers
a.
To find the equations that define the upper and lower semi ellipses as function of x .
a.
Answer to Problem 56E
The equations that define the upper and lower semi ellipses as function of x is
Explanation of Solution
Given information:
The given equation of ellipse is
Calculation:
Solve the equation for y .
Therefore,
The equations that define the upper and lower semi ellipses as function of x is
b.
To write an integral expression that gives the area of the ellipse.
b.
Answer to Problem 56E
The integral expression for the area of ellipse is
Explanation of Solution
Given information:
The given equation of ellipse is
Calculation:
Since the ellipse is symmetric abut x axis and centred at the origin. Therefore area of half ellipse can be calculated as integrating y as a function of x in the range of major axis that is
And for the area of ellipse multiply the area of half ellipse by 2, as the ellipse is symmetric.
Therefore,
The integral expression for the area of ellipse is
c.
To find the area of ellipses for various lengths of a and b .
c.
Answer to Problem 56E
For
For
Similarly for different values of a and b area of the ellipses can be evaluated.
Explanation of Solution
Given information:
The given equation of ellipse is
Calculation:
Consider
Then the area of the ellipse centred at the origin.
Again consider
Then the area of the ellipse centred at the origin.
Similarly for different values of a and b area of the ellipses can be evaluated.
d.
To write the form of areas found in above parts.
d.
Answer to Problem 56E
The form of area is
Explanation of Solution
Given information:
The given equation of ellipse is
Calculation:
First write the integral obtained for calculation of area of ellipse.
Now integrate above expression.
Therefore,
The form of area is
e.
To proof that the simple area formulae of ellipse is same as the integral formula of area of the ellipse.
e.
Answer to Problem 56E
The graph of
Explanation of Solution
Given information:
The function is
Proof:
The simple formula of area of ellipse centred at origin is
First write the integral obtained for calculation of area of ellipse.
Now integrate above expression.
Therefore,
The integral area formula gives the exact value of area as the simple formula of area of ellipse.
Chapter 8 Solutions
Calculus: Graphical, Numerical, Algebraic: Solutions Manual
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