Comprehensive Calculus Notes: Volumes of Revolution via Washers and Cylindrical Shells

The volume of a solid generated by rotating a region between two curves about a horizontal axis is calculated using the Washer Method if the cross-sectional area is a circle with a hole (a washer).
The basic formula for the Washer Method is based on the difference between the volumes of a large cylinder and a small cylinder: $V = \pi \int_a^b [R(x)^2 - r(x)^2] \,dx$
Where:
- $R(x)$ is the Outer Radius (distance from the axis of rotation to the outer curve).
- $r(x)$ is the Inner Radius (distance from the axis of rotation to the inner curve).

Region Definition: The region is bounded by $y = 1 + \sec(x)$ , $y = 3$ , and the vertical boundaries at $x = -\pi/3$ and $x = \pi/3$ .
Axis of Rotation: The line $y = 1$ .
Step 1: Finding Intersection Points:
- Set the functions equal to each other to find where the curves meet: $3 = 1 + \sec(x)$ $\sec(x) = 2$ $\frac{1}{\cos(x)} = 2 \implies \cos(x) = \frac{1}{2}$
- Between $x = -\pi/2$ and $x = \pi/2$ , the cosine equals $1/2$ at $x = \frac{\pi}{3}$ and $x = -\frac{\pi}{3}$ .
Step 2: Defining Radii:
- The distance from the axis of rotation $y = 1$ must be calculated by subtracting the axis from the function values ( $y_{\text{upper}} - y_{\text{lower}}$ ).
- Large Radius ( $R$ ): The distance from the axis $y = 1$ to the farthest curve $y = 3$ . $R = 3 - 1 = 2$
- Small Radius ( $r$ ): The distance from the axis $y = 1$ to the inner curve $y = 1 + \sec(x)$ . $r = (1 + \sec(x)) - 1 = \sec(x)$
Step 3: Setting Up and Evaluating the Integral:
- Using the symmetry of the secant squared function, the integral can be calculated from $0$ to $\pi/3$ and then doubled: $V = \pi \int_{-\pi/3}^{\pi/3} [2^2 - (\sec(x))^2] \,dx$ $V = 2\pi \int_0^{\pi/3} [4 - \sec^2(x)] \,dx$
- The antiderivative of $4$ is $4x$ , and the antiderivative of $\sec^2(x)$ is $\tan(x)$ . $V = 2\pi [4x - \tan(x)]|_0^{\pi/3}$ $V = 2\pi [4(\pi/3) - \tan(\pi/3)] - 2\pi [0 - 0]$ $V = 2\pi [\frac{4\pi}{3} - \sqrt{3}]$

Example Comparison: Finding the volume for a region bounded by $y = \arctan(x^2)$ and $y = 1 + xe^{-x^3}$ rotated about the x-axis.
Essential Settings: The calculator must be in Radian Mode for all trigonometric calculus problems.
Finding Intersections:
- Use the Second + CALC menu and select 5: Intersect.
- Choose the first curve, the second curve, and provide a guess near the intersection point.
- Approximate intersection values for this specific problem are given as $x \approx -0.5057$ and $x \approx 1.3915$ .
Setup:
- Large radius: $R(x) = 1 + xe^{-x^3}$
- Small radius: $r(x) = \arctan(x^2)$
- $V = \pi \int_{-0.5057}^{1.3915} [(1 + xe^{-x^3})^2 - (\arctan(x^2))^2] \,dx$
Calculation Tips:
- Use the MATH + 9 (fnInt) function to integrate.
- Syntax: fnInt(integrand, variable, lower limit, upper limit).
- To edit mistakes in long expressions, use the Insert (Second + DEL) feature to add characters without overwriting.
- Multiplication by $\pi$ can be done at the end to minimize rounding errors during calculation stages.

Purpose of the Method: Cylindrical shells provide an alternative when solving for variables (e.g., solving $y = 2x^2 - x^3$ for $x$ ) is difficult or impossible, making Disk/Washer methods impractical for rotations about the y-axis.
Conceptual Derivation: Imagine nesting thin cans (shells) inside one another.
- The volume of a single shell is roughly the circumference times height times thickness: $V = 2\pi \times \text{radius} \times \text{height} \times \text{thickness}$
Integral Formula:
- For rotation about a vertical axis (like the y-axis): $V = \int_a^b 2\pi x f(x) \,dx$
- Radius: Usually the distance from the shell to the axis of rotation ( $x - \text{axis}$ or $\text{axis} - x$ ).
- Height: The difference between the upper and lower functions ( $y_{\text{upper}} - y_{\text{lower}}$ ).
- Thickness: $dx$ or $dy$ .
Axis of Rotation vs. Shell Orientation:
- If rotating about a vertical axis, use vertical shells (integrate with respect to $x$ ).
- If rotating about a horizontal axis, use horizontal shells (integrate with respect to $y$ ).

Default Strategy (Decision Matrix):
- Vertical Axis + Equation solved for y (y = f(x)): Use Cylindrical Shells (integrates with respect to $x$ ).
- Horizontal Axis + Equation solved for y (y = f(x)): Use Disks/Washers (integrates with respect to $x$ ).
- Horizontal Axis + Equation solved for x (x = g(y)): Use Cylindrical Shells (integrates with respect to $y$ ).
- Vertical Axis + Equation solved for x (x = g(y)): Use Disks/Washers (integrates with respect to $y$ ).
Practical Considerations:
- Check if the boundaries change. If using one method requires splitting the area into two integrals because the "top" and "bottom" curves switch, it may be easier to switch to the other method for a single integral solution.
- Always simplify the integrand (e.g., multiply through by the radius) before integrating.

Case 1: Rotating the region bounded by $y = 4x - x^2$ and $y = x$ about the vertical line $x = 7$ .
- Radius: $7 - x$ (distance from rightmost axis to leftmost point).
- Height: $(4x - x^2) - x = 3x - x^2$ .
- Limits: Intersection at $x = 0$ and $x = 3$ .
- Integral: $V = 2\pi \int_0^3 (7-x)(3x-x^2) \,dx$ .
Case 2: Rotating the region bounded by $x = 3y - y^2$ and $x = 2$ about the horizontal line $y = 3$ .
- Method: Cylindrical Shells (horizontal shells for a horizontal axis).
- Radius: $3 - y$ .
- Height: $(3y - y^2) - 2$ .
- Limits: Intersection points found via $y^2 - 3y + 2 = 0 \implies y = 1, 2$ .
- Integral: $V = 2\pi \int_1^2 (3-y)(3y - y^2 - 2) \,dy$ .

Prompt: "How much are you on rounding error?"
Response: The professor emphasizes that setting up the integral correctly is worth the majority of points (e.g., 8 out of 10). If the final decimal is slightly off due to rounding limits (e.g., using 1.37 instead of 1.39), it might only result in a small point deduction. If the answer is wildly different, it usually indicates a conceptual setup error.
Prompt: "Don't have the same answer as the book?"
Response: The instructor noted that some textbook problems (like the secant one) originally lacked bounds, which would lead to an infinite volume. Always ensure you are using the correct parameters and bounds provided in the specific version of the problem you are solving.
Prompt: "How do you add a space on the calculator?"
Response: Use the Insert key (Second + DEL). This allows you to add variables or numbers into the middle of a string without deleting what is already typed.
Prompt: "Why is the radius $3 - y$ ?"
Response: Because the rotation axis is at $y = 3$ . To find the length of the radius (which must be positive), you subtract the lower y-value from the upper y-value: $y_{\text{axis}} - y_{\text{shell}} = 3 - y$ .