Math Problem Statement

Consider the region bounded by y=x^3, y=0, and x=1. Find the volume of the solid obtained by rotating the region around the vertical line x=2

Solution

We are tasked with finding the volume of the solid formed by rotating the region bounded by y=x3y = x^3, y=0y = 0, and x=1x = 1 around the vertical line x=2x = 2. To solve this, we will use the method of cylindrical shells.

Step-by-Step Solution:

  1. General Setup for Cylindrical Shells: When rotating a region around a vertical line not passing through the region itself, the volume can be found using the formula for the volume of cylindrical shells:

    V=2πab(radius)(height)dxV = 2\pi \int_{a}^{b} \left( \text{radius} \right) \cdot \left( \text{height} \right) \, dx

    Here:

    • The radius of a typical shell is the distance from xx to the line x=2x = 2, which is 2x2 - x.
    • The height of a typical shell is given by the function y=x3y = x^3.
  2. Define the Limits of Integration: The region is bounded by x=0x = 0 and x=1x = 1, so the limits of integration are from 00 to 11.

  3. Set Up the Integral: Using the formula for the volume of cylindrical shells, we have:

    V=2π01(2x)(x3)dxV = 2\pi \int_{0}^{1} (2 - x)(x^3) \, dx

  4. Simplify the Integrand: Expand the expression inside the integral:

    V=2π01(2x3x4)dxV = 2\pi \int_{0}^{1} \left( 2x^3 - x^4 \right) \, dx

  5. Integrate: Now integrate term by term:

    V=2π[2x44x55]01V = 2\pi \left[ \frac{2x^4}{4} - \frac{x^5}{5} \right]_{0}^{1}

    Simplify the expressions inside the brackets:

    V=2π[1215]V = 2\pi \left[ \frac{1}{2} - \frac{1}{5} \right]

  6. Compute the Final Value: Simplify the expression:

    V=2π(510210)=2π310=6π10=3π5V = 2\pi \left( \frac{5}{10} - \frac{2}{10} \right) = 2\pi \cdot \frac{3}{10} = \frac{6\pi}{10} = \frac{3\pi}{5}

Thus, the volume of the solid is:

3π5\boxed{\frac{3\pi}{5}}

Would you like further clarification or a more detailed breakdown of any of the steps?

Related Questions:

  1. How would the solution change if the rotation was around the line x=3x = 3?
  2. Can we use the disk/washer method instead of cylindrical shells for this problem?
  3. What if the region were rotated around the horizontal line y=1y = -1?
  4. How do we approach this problem if the region is instead bounded by y=x2y = x^2?
  5. How do we apply the cylindrical shell method when rotating around x=1x = -1?

Tip: When using the method of cylindrical shells, the radius is always the distance from the rotation axis to the variable xx, which must be correctly defined depending on the position of the axis.

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Math Problem Analysis

Mathematical Concepts

Calculus
Solid of Revolution
Cylindrical Shell Method

Formulas

V = 2π ∫[a, b] (radius) * (height) dx
Radius = 2 - x
Height = y = x^3

Theorems

Cylindrical Shell Method

Suitable Grade Level

Grades 11-12