Consider the problem of designing cylindrical tanks for liquid storage. A region is bounded by a curve and the x-axis, and revolving this region generates the tank’s volume. Address the following multi-part problem:
(a) Discuss the fundamental differences in setting up integrals to calculate volumes when the region is revolved about the x-axis versus the y-axis. Explain the applicability of the disk/washer method for x-axis rotation and the shell method for y-axis rotation, emphasizing how the integral expressions differ in terms of the variable of integration and the function used.
(b) For a specific tank design, the region is defined by the curve \(y = \sqrt{x}\), the x-axis, and the vertical line \(x = 4\). Compute numerically:
– The volume of the tank formed when this region is revolved about the x-axis using the disk method.
– The volume of the tank formed when the same region is revolved about the y-axis using the shell method.
(c) Based on your integral set-ups and numerical evaluations in parts (a) and (b), explain how the choice of the axis of revolution impacts the resulting volume and the complexity of the calculation. In a real-world design context where minimizing material is crucial, discuss which axis of revolution might be preferred for tank design and why, considering both volume and surface area implications (assume surface area is related to the method of revolution).
Question
Discuss the fundamental differences in setting up integrals for volumes of revolution about the x-axis compared to the y-axis. Explain how the choice of method (disk/washer vs shell) affects the integral expression, focusing on the variable of integration and the function used.
Hint
Hint
Hint
Consider the orientation of slices (perpendicular or parallel to the axis of rotation) and how this dictates the variable of integration and the form of the integrand for disk/washer and shell methods.
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Complete Answer
For x-axis rotation, the disk/washer method uses vertical slices (perpendicular to the x-axis), integrating with respect to x, where the radius is given by the function y = f(x). The volume element is \(\pi [f(x)]^2 dx\). For y-axis rotation, using the shell method with vertical slices (parallel to the y-axis) integrates with respect to x, where the radius is x and the height is f(x). The volume element is \(2\pi x f(x) dx\). The disk/washer method for y-axis rotation and shell method for x-axis rotation are also possible but often less straightforward when y is given as a function of x.
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Question
For the region bounded by \(y = \sqrt{x}\), the x-axis, and \(x = 4\), compute numerically:
– The volume of the tank formed by revolving this region about the x-axis using the disk method.
– The volume of the tank formed by revolving the same region about the y-axis using the shell method.
Hint
Hint
Hint
Set up the definite integrals using the disk method for x-axis rotation and the shell method for y-axis rotation, with limits of integration from x = 0 to x = 4. Use \(V = \pi \int [f(x)]^2 dx\) for disks and \(V = 2\pi \int x f(x) dx\) for shells.
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Volume about the x-axis: \(8\pi\). Volume about the y-axis: \(\frac{128\pi}{5}\).
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Based on the integral set-ups and numerical evaluations in parts (a) and (b), explain how the choice of axis of revolution influences the volume calculation. In a real-world tank design context focused on minimizing material, discuss whether revolving about the x-axis or y-axis would be more efficient for creating a tank, considering the calculated volumes and potential surface area differences.
Hint
Hint
Hint
Compare the volumes obtained in part (b) and consider how the shape of the tank differs based on the axis of revolution. Think about which shape might enclose a larger volume with less surface area, relating to material efficiency.
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Revolving about different axes results in different volumes and tank shapes. In part (b), the volume about the y-axis (\(\frac{128\pi}{5}\)) is greater than about the x-axis (\(8\pi\)). For minimizing material in tank design, we typically aim to maximize volume for a given surface area (or minimize surface area for a given volume). While this question only calculates volume, the choice of axis significantly alters the tank’s proportions (tall and narrow vs. short and wide), which affects surface area and thus material usage. In this case, if volume maximization is prioritized, y-axis rotation yields a larger volume. However, a full optimization would require considering surface area as well, which is influenced by the axis of revolution.
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