How is the radius calculated when rotating a function $f(x)$ around a horizontal line $y = k$?

The radius is the absolute difference between the function and the axis, expressed as $r = |f(x) - k|$. This represents the vertical distance from the curve to the line of rotation.

When rotating around the line $x = 3$, what must be done to the boundary equations $y = f(x)$?

The equations must be solved for $x$ in terms of $y$ (e.g., $x = g(y)$). Since the rotation is around a vertical axis, the integration must be performed with respect to $y$ ($dy$).

What is the difference between the Disk Method and the Washer Method when rotating around a line $y = k$?

The Disk Method is used when the region is bounded by the axis of rotation itself, resulting in a solid cross-section. The Washer Method is used when there is a gap between the region and the axis, resulting in a hollow cross-section with two radii.

If the axis of rotation is $y = 10$ and the region lies between $y = 2$ and $y = 5$, which curve provides the outer radius $R$?

The curve $y = 2$ provides the outer radius because it is further from the axis $y = 10$ (distance of 8) than the curve $y = 5$ (distance of 5).

Why is the formula $\pi \int (f(x) - g(x) - k)^2 dx$ incorrect for the washer method?

This formula squares the total height of the region after a shift, rather than subtracting the volume of the inner hole from the outer solid. The correct approach is to subtract the squares of the individual radii: $\pi \int (R^2 - r^2) dx$.

How do you determine the limits of integration when rotating around a vertical line $x = k$?

The limits are the $y$-coordinates of the intersection points of the boundary curves. This is because the integration is performed along the $y$-axis ($dy$) to sum the horizontal washer slices.

What happens to the volume calculation if you accidentally swap the outer radius $R$ and inner radius $r$ in the formula?

The resulting volume will be the correct magnitude but will have a negative sign. Volume must always be positive, so a negative result usually indicates that the radii were swapped or the bounds were reversed.

In the context of 'Other Axes', what does the term 'Outer Radius' specifically refer to?

It refers to the distance from the axis of rotation to the boundary of the region that is most distant. It is the radius of the larger disk from which the smaller 'hole' disk is subtracted.

When rotating around $y = -2$, how does the radius calculation change compared to rotating around the x-axis ($y=0$)?

Instead of the radius being $f(x)$, it becomes $f(x) - (-2)$, which simplifies to $f(x) + 2$. This reflects the increased distance from the curve to the axis located below the x-axis.

What is the geometric shape of the 'slice' used in the Washer Method?

The slice is a thin right circular cylinder with a smaller concentric right circular cylinder removed from its center. Its volume is the difference between the volumes of the two cylinders.

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Unit 8: Applications of Integration

Washer Method Around Other Axes

Summary

The Washer Method Around Other Axes extends the standard volume of revolution technique to cases where the axis of rotation is a line other than the x or y axes. By calculating the distance between the boundary functions and the specific line of rotation, we define outer and inner radii that account for the 'hollow' center of the resulting solid.

1. Definition & Core Concepts

Washer Method: A technique for finding the volume of a solid of revolution with a hollow center, created by rotating a region between two curves around an axis.
Axis of Rotation ( $k$ ): A horizontal line ( $y = k$ ) or vertical line ( $x = k$ ) that serves as the center of the rotation, which is not one of the coordinate axes.
Outer Radius ( $R$ ): The distance from the axis of rotation to the boundary curve that is geometrically further away from that axis.
Inner Radius ( $r$ ): The distance from the axis of rotation to the boundary curve that is geometrically closer to that axis.

Diagram showing a region between two curves being rotated around a horizontal line y=k. The outer radius R(x) and inner radius r(x) are shown as vertical distances from the line y=k to the curves.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Washer Method Around Other Axes

Summary

1. Definition & Core Concepts

Washer Method: A technique for finding the volume of a solid of revolution with a hollow center, created by rotating a region between two curves around an axis.
Axis of Rotation ( $k$ ): A horizontal line ( $y = k$ ) or vertical line ( $x = k$ ) that serves as the center of the rotation, which is not one of the coordinate axes.
Outer Radius ( $R$ ): The distance from the axis of rotation to the boundary curve that is geometrically further away from that axis.
Inner Radius ( $r$ ): The distance from the axis of rotation to the boundary curve that is geometrically closer to that axis.

Diagram showing a region between two curves being rotated around a horizontal line y=k. The outer radius R(x) and inner radius r(x) are shown as vertical distances from the line y=k to the curves.

2. Underlying Principles

Radius as Distance: In the standard method, the radius is simply the function value $f(x)$ . When the axis shifts to $y=k$ , the radius becomes the absolute difference $|f(x) - k|$ .
Volume of a Washer: The volume of an infinitesimal slice is the area of the outer circle minus the area of the inner circle, multiplied by the thickness: $dV = \pi(R^2 - r^2) dx$ .
Summation via Integration: The definite integral sums these infinite thin washers across the interval $[a, b]$ to find the total volume of the solid.

3. Methods & Techniques

Horizontal Axis of Rotation ( $y = k$ )

Identify Radii: Determine which function is further ( $f_{outer}$ ) and which is closer ( $f_{inner}$ ) to the line $y=k$ .
Set up Integral: Use the formula with respect to $x$ :

$V = \pi \int_a^b ([f_{outer}(x) - k]^2 - [f_{inner}(x) - k]^2) dx$

Vertical Axis of Rotation ( $x = k$ )

Rewrite Functions: Ensure all boundary equations are solved for $x$ in terms of $y$ (e.g., $x = g(y)$ ).
Set up Integral: Use the formula with respect to $y$ :

$V = \pi \int_c^d ([g_{outer}(y) - k]^2 - [g_{inner}(y) - k]^2) dy$

4. Key Distinctions

Feature	Horizontal Axis ( $y=k$ )	Vertical Axis ( $x=k$ )
Integration Variable	$dx$	$dy$
Radius Calculation	Vertical distance ( $y$ -values)	Horizontal distance ( $x$ -values)
Function Form	$y = f(x)$	$x = g(y)$
Bounds	$x$ -intercepts or vertical lines	$y$ -intercepts or horizontal lines

Disk vs. Washer: Use the Disk method if the region is flush against the axis of rotation; use the Washer method if there is a gap between the region and the axis.

5. Exam Strategy & Tips

Sketch the Axis: Always draw the line of rotation first. It is easy to confuse a 'top' function with an 'outer' radius if you don't visualize the axis position.
Check the Order: While squaring makes the sign of $(f(x)-k)$ irrelevant, correctly identifying the outer radius is crucial for the subtraction order $(R^2 - r^2)$ to ensure a positive volume.
Bounds Consistency: If you are integrating with respect to $y$ , ensure your limits of integration are the $y$ -coordinates of the intersection points, not the $x$ -coordinates.

6. Common Pitfalls & Misconceptions

The Square of the Difference: A frequent error is writing $(f(x) - g(x) - k)^2$ . You must square the individual radii separately: $(f(x)-k)^2 - (g(x)-k)^2$ .
Ignoring the Shift: Students often forget to subtract $k$ from the function, essentially calculating the volume as if it were rotated around the standard $x$ or $y$ axis.
Axis Position: If the axis $y=k$ is above the region, the distance is $k - f(x)$ . If the axis is below, it is $f(x) - k$ . Both yield the same result when squared, but the logic helps in identifying $R$ vs $r$ .

Radius as Distance: In the standard method, the radius is simply the function value $f(x)$ . When the axis shifts to $y=k$ , the radius becomes the absolute difference $|f(x) - k|$ .
Volume of a Washer: The volume of an infinitesimal slice is the area of the outer circle minus the area of the inner circle, multiplied by the thickness: $dV = \pi(R^2 - r^2) dx$ .
Summation via Integration: The definite integral sums these infinite thin washers across the interval $[a, b]$ to find the total volume of the solid.

Horizontal Axis of Rotation ( $y = k$ )

Identify Radii: Determine which function is further ( $f_{outer}$ ) and which is closer ( $f_{inner}$ ) to the line $y=k$ .
Set up Integral: Use the formula with respect to $x$ :

$V = \pi \int_a^b ([f_{outer}(x) - k]^2 - [f_{inner}(x) - k]^2) dx$

Vertical Axis of Rotation ( $x = k$ )

Rewrite Functions: Ensure all boundary equations are solved for $x$ in terms of $y$ (e.g., $x = g(y)$ ).
Set up Integral: Use the formula with respect to $y$ :

$V = \pi \int_c^d ([g_{outer}(y) - k]^2 - [g_{inner}(y) - k]^2) dy$

Feature	Horizontal Axis ( $y=k$ )	Vertical Axis ( $x=k$ )
Integration Variable	$dx$	$dy$
Radius Calculation	Vertical distance ( $y$ -values)	Horizontal distance ( $x$ -values)
Function Form	$y = f(x)$	$x = g(y)$
Bounds	$x$ -intercepts or vertical lines	$y$ -intercepts or horizontal lines

Disk vs. Washer: Use the Disk method if the region is flush against the axis of rotation; use the Washer method if there is a gap between the region and the axis.

Sketch the Axis: Always draw the line of rotation first. It is easy to confuse a 'top' function with an 'outer' radius if you don't visualize the axis position.
Check the Order: While squaring makes the sign of $(f(x)-k)$ irrelevant, correctly identifying the outer radius is crucial for the subtraction order $(R^2 - r^2)$ to ensure a positive volume.
Bounds Consistency: If you are integrating with respect to $y$ , ensure your limits of integration are the $y$ -coordinates of the intersection points, not the $x$ -coordinates.

The Square of the Difference: A frequent error is writing $(f(x) - g(x) - k)^2$ . You must square the individual radii separately: $(f(x)-k)^2 - (g(x)-k)^2$ .
Ignoring the Shift: Students often forget to subtract $k$ from the function, essentially calculating the volume as if it were rotated around the standard $x$ or $y$ axis.
Axis Position: If the axis $y=k$ is above the region, the distance is $k - f(x)$ . If the axis is below, it is $f(x) - k$ . Both yield the same result when squared, but the logic helps in identifying $R$ vs $r$ .

Washer Method Around Other Axes

Summary

1. Definition & Core Concepts

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Washer Method Around Other Axes

Summary

1. Definition & Core Concepts

2. Underlying Principles

3. Methods & Techniques

Horizontal Axis of Rotation (y=ky = ky=k)

Vertical Axis of Rotation (x=kx = kx=k)

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Horizontal Axis of Rotation (y=ky = ky=k)

Vertical Axis of Rotation (x=kx = kx=k)

Horizontal Axis of Rotation ( $y = k$ )

Vertical Axis of Rotation ( $x = k$ )

Horizontal Axis of Rotation ( $y = k$ )

Vertical Axis of Rotation ( $x = k$ )