Problem 16 Let a curve \(y=f(x)\) be revolv... [FREE SOLUTION] (2024)

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A surface of revolution is created when we rotate a curve described by the function, such as \( y = f(x) \), around an axis; in this problem, it's around the \( x \)-axis.

If you look at the cross sections of this surface through a plane where \( x \) is a constant value, say \( x = c \), then you end up slicing the surface parallel to the \( yz \)-plane.
This means you're essentially taking a 'snapshot' of the surface at that point.
Every cross-section in these conditions forms a circle.
The radius of this circle is given by the function \( f(x) \) at the point \( x = c \).

So if our curve \( y = f(c) \), then at that constant cross-sectional value, the radius of the circle is exactly \( f(c) \).
This comes from the nature of the surface being revolved, which stretches out directly from the axis at the height or depth defined by \( f(c) \).
To recap:

• The plane \( x = c \) slices the surface parallel to the \( yz \)-plane.
• Each cross section forms a circle whose radius is \( f(c) \).

Understanding the cross section helps us write the general equation for a surface of revolution.

Given that each cross section in the plane \( x = c \) forms a circle of radius \( f(c) \), we can use the equation for a circle.
A circle centered on the origin in the \( yz \)-plane with radius \( f(c) \) has the equation: \[ y^2 + z^2 = f(c)^2 \].
But remember that we're dealing with a surface, not just a single cross-sectional slice.
The variable \( x \) can take any value along the curve, making \( f \) a function of \( x \), rather than just a number.
Therefore, the general equation encapsulates the fact that \( y \) and \( z \) form circles of varying radii, depending on \( x \).

So the general equation of the surface of revolution is: \[ y^2 + z^2 = [f(x)]^2 \].

• This shows how the shape bends and forms as \( x \) changes.
• For every \( x \) value, the distance from the \( x \) axis defines the circle's radius.

To verify this general equation, let's use a specific function.

Consider the function \( f(x) = x^2 \).
Substitute this function into our general surface of revolution equation: \[ y^2 + z^2 = [x^2]^2 \ = x^4 \].
This represents the surface where each cross section, for a specific \( x \) value, forms a circle with a radius equal to \( x^2 \).

For example:

• When \( x = 1 \), the radius is \(1^2 = 1\).
• When \( x = 2 \), the radius is \(2^2 = 4\).

This approach confirms that our general equation accurately represents the surface of revolution where \( f(x) = x^2 \).

To summarize:

• We applied a specific function \( f(x) \) to the general equation.
• Each slice (cross section) matched the theoretical expectation (circle of radius \( x^2 \)).