Math Problem Statement

Find the volume of the solid obtained by rotating about the y-axis the region bounded by y = 3x^2 - x^3 and y = 0.

Solution

To solve this problem, we need to use the shell method for finding the volume of the solid. Let's solve each part step by step:

Part (a)

The formula for the shell method involves the radius, xx, and the height, f(x)f(x). From the problem:

  • The height of the shell is given by the function f(x)=3x2x3f(x) = 3x^2 - x^3, which is the boundary of the region.

So, the height f(x)=3x2x3f(x) = 3x^2 - x^3.


Part (b)

The volume VV using the shell method is given by: V=2π0axf(x)dxV = 2\pi \int_0^a x \cdot f(x) \, dx where f(x)=3x2x3f(x) = 3x^2 - x^3, and aa is the upper limit of integration (where y=3x2x3y = 3x^2 - x^3 intersects y=0y = 0).

To find aa, solve 3x2x3=03x^2 - x^3 = 0: x2(3x)=0x^2(3 - x) = 0 Thus, x=0x = 0 or x=3x = 3. So, a=3a = 3.

The volume formula becomes: V=2π03x(3x2x3)dxV = 2\pi \int_0^3 x (3x^2 - x^3) \, dx


Part (c)

Simplify the integrand: x(3x2x3)=3x3x4x (3x^2 - x^3) = 3x^3 - x^4 So: V=2π03(3x3x4)dxV = 2\pi \int_0^3 (3x^3 - x^4) \, dx

Now compute the integral: 03(3x3x4)dx=[3x44x55]03\int_0^3 (3x^3 - x^4) \, dx = \left[ \frac{3x^4}{4} - \frac{x^5}{5} \right]_0^3

Evaluate at the limits: At x=3x = 3: 3(34)4(35)5=3(81)42435=24342435\frac{3(3^4)}{4} - \frac{(3^5)}{5} = \frac{3(81)}{4} - \frac{243}{5} = \frac{243}{4} - \frac{243}{5}

Find a common denominator (20): 2434=121520,2435=97220\frac{243}{4} = \frac{1215}{20}, \quad \frac{243}{5} = \frac{972}{20} 24342435=121597220=24320\frac{243}{4} - \frac{243}{5} = \frac{1215 - 972}{20} = \frac{243}{20}

So: V=2π24320=486π20=243π10V = 2\pi \cdot \frac{243}{20} = \frac{486\pi}{20} = \frac{243\pi}{10}


Final Answer:

V=243π10V = \frac{243\pi}{10}

Let me know if you have any questions or want further details!

Questions to explore:

  1. How does the shell method differ from the disk/washer method?
  2. Why is f(x)=3x2x3f(x) = 3x^2 - x^3 used for height in this problem?
  3. Can the same volume be computed using the disk/washer method? If so, how?
  4. What role does symmetry play in simplifying the integration?
  5. How would the integral change if the region was rotated about a different axis?

Tip: Always double-check the limits of integration when solving problems involving revolutions. Misidentifying limits can lead to significant errors.

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Math Problem Analysis

Mathematical Concepts

Volume of solids of revolution
Shell method
Definite integration

Formulas

Shell method formula: V = 2π ∫[a to b] x f(x) dx

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12