Question

Areas of Surfaces of RevolutionIn Exercises 23–26, find the areas of the surfaces generated by revolving the curves about the given axes.                                                                                                                                                                                                                                                                                                                                _______                                                                                                                                                                                                                                                                          y = √4y ― y²  , 1 ≤ y ≤ 2 ;  y-axis

Accepted Answer

First, recognize that the problem involves finding the surface area generated by revolving a curve around the y-axis. The formula for the surface area when revolving around the y-axis is:

$$ S = \int_{a}^{b} 2\pi x(y) \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy $$

where $$x is $$expressed as a function of $$y. $$Given the curve $$y = \sqrt{4y - y^2$}$$, the first step is to rewrite the curve to express $$x as a $$function of $$y. $$Notice that the equation looks like it might represent $$x in $$terms of $$y$$, so isolate $$x$$ accordingly. Since the curve is given as $$y = \sqrt{4y - y^2$}$$, check if this is a typo or if it should be $$x = \sqrt{4y - y^2$} to $$revolve around the y-axis. Assuming the curve is $$x = \sqrt{4y - y^2$}$$ for $$1 \leq y \leq 2$$, compute the derivative $$\frac{dx}{dy}$$ using the chain rule:

$$ \frac{dx}{dy} = \frac{d}{dy} \left( \sqrt{4y - y^2} \right) = \frac{1}{2\sqrt{4y - y^2}} (4 - 2y) $$ Substitute $$x(y)$$ and $$\frac{dx}{dy}$$ into the surface area integral formula:

$$ S = \int_{1}^{2} 2\pi \sqrt{4y - y^2} \sqrt{1 + \left( \frac{4 - 2y}{2\sqrt{4y - y^2}} \right)^2} \, dy $$ Simplify the expression inside the square root and the integral as much as possible to make the integral easier to evaluate. Then, set up the integral for evaluation either by hand or using a computational tool to find the surface area.

Areas of Surfaces of Revolution
In Exercises 23–26, find the areas of the surfaces generated by revolving the curves about the given axes.
_______
y = √4y ― y² , 1 ≤ y ≤ 2 ; y-axis

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Key Concepts

Surface Area of Revolution

Parametrization and Variable of Integration

Arc Length Element (ds)