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Chapter 6: Problem 19

\(15-20\) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell. $$ y=x^{3}, y=0, x=1 ; \quad \text { about } y=1 $$

Short Answer

Expert verified
The volume is \(\frac{3\pi}{5}\).

Step by step solution

01

Sketch the Region and Identify Bounds

The region is bounded by the curves \(y = x^{3}\), \(y = 0\), and \(x = 1\). The rotation axis is \(y = 1\). Sketching this involves plotting \(y = x^{3}\), a cubic curve, from \(x = 0\) to \(x = 1\), and the horizontal line \(y = 0\), giving a region below the cubic curve and above the x-axis, bounded vertically at \(x = 1\). Visualize this rotating around \(y = 1\).
02

Set up the Cylindrical Shell Formula

The formula for the volume using cylindrical shells is given by: \[ V = 2\pi \int_{a}^{b} (radius)(height) \, dx \] where the radius is the distance from the function to the axis of rotation \(\left|y=1-x^3\right|\) and the height is \(x\).
03

Determine the Radius and Height

Since we are rotating around \(y = 1\), the distance from a point on the function \((x, x^3)\) to \(y = 1\) is \(1 - x^3\) (the upper blade) for the radius and the height of the shell is given by \(x\).
04

Integrate Using the Shell Method

Substitute into the formula: \[ V = 2\pi \int_{0}^{1} (1 - x^3)(x) \, dx = 2\pi \int_{0}^{1} (x - x^4) \, dx \]Let's solve the integral:\[ V = 2\pi \left[ \frac{x^2}{2} - \frac{x^5}{5} \right]_{0}^{1} \]Evaluating the definite integral from 0 to 1:\[ V = 2\pi \left( \frac{1}{2} - \frac{1}{5} \right) = 2\pi \left( \frac{5}{10} - \frac{2}{10} \right) = 2\pi \cdot \frac{3}{10} = \frac{3\pi}{5} \].
05

Finalize the Volume Calculation

Completing the integration process, we find that the volume of the solid obtained by rotating the region about \(y=1\) is \(\frac{3\pi}{5}\). This confirms our calculation with cylindrical shells.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume by Rotation
When you think about volume by rotation, imagine transforming a flat shape into a 3D object by turning it around a line. This method is commonly used in mathematics to determine the volume of solids generated from revolving a region or shape around an axis. For this exercise, we used a method called cylindrical shells.
Rotation gives depth to objects, allowing us to measure their volume.
Here are key points for understanding this concept:
  • A flat shape, like the one bound by curves, revolves around a fixed line (axis of rotation).
  • The rotation creates a 3D solid with depth.
  • We calculate the volume of this solid through integration techniques such as cylindrical shells.
This process involves calculating the area of infinitely thin cylindrical shells that form the solid. Adding up the volumes of these shells yields the total volume of the solid.
Cubic Curve
A cubic curve is a curve described by a polynomial equation of degree 3. In this problem, the cubic curve given is represented by the equation: \( y = x^3 \).
The cubic curve has a distinct shape, starting at the origin and increasing at an accelerating rate as x increases, often creating a characteristic swooping up shape. Understanding this involves:
  • Identifying the bounds: we work with x between 0 and 1.
  • Recognizing the curve's behavior: \( y = x^3 \) rises steeply from zero as x increases.
  • The curve is an essential part of the region whose volume we're calculating.
Visualizing the curve helps in sketching the region to be revolved.
Definite Integral
The definite integral is a fundamental concept in calculus, used to find the area under a curve or the total change of a function. In this exercise, the definite integral calculates the volume of the solid via the shell method.
  • The integral expression we used is \( V = 2\pi \int_{0}^{1} (1 - x^3)x \, dx \).
  • This integral measures the accumulated area of thin shells from \(x = 0\) to \(x = 1\).
  • By solving this integral, we capture the total volume of the solid created.
Essentially, the definite integral sums up all these tiny volumes to give us the final answer.
Axis of Rotation
The axis of rotation is a fixed, straight line around which a shape revolves to generate a 3D object. In this problem, the axis is given by \(y=1\). Understanding the axis of rotation is crucial because:
  • It serves as a pivot point for rotation.
  • The distance from the axis to a point on the shape becomes the radius for the cylindrical shell.
  • For our specific example, the distance from \(y=1\) to any point \( (x, x^3) \) is the radius \( 1 - x^3 \).
This axis orientation significantly affects the calculations and the resulting shape of the solid by rotation.

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Most popular questions from this chapter

\(39-42\) Each integral represents the volume of a solid. Describe the solid. $$\pi \int _ { 0 } ^ { 1 } \left( y ^ { 4 } - y ^ { 8 } \right) d y$$

\(21-26\) Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $$ y=1 /\left(1+x^{2}\right), y=0, x=0, x=2 ; \quad \text { about } x=2 $$

Find the work done if a constant force of 100 \(\mathrm{lb}\) is used to pull a cart a distance of 200 \(\mathrm{ft} .\)

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