Math Problem Statement

Find the volume formed by rotating the region enclosed by:

x=4y, y^3=x with y≥0

about the y-axis

volume=

Solution

We are tasked with finding the volume of the solid formed by rotating the region enclosed by the curves x=4yx = 4y and y3=xy^3 = x, where y0y \geq 0, about the y-axis.

Step 1: Determine the points of intersection

To find the region, we first need to determine where the two curves intersect by setting x=4yx = 4y equal to x=y3x = y^3: 4y=y34y = y^3 Dividing both sides by yy (since y0y \geq 0 excludes y=0y = 0): 4=y24 = y^2 Thus, y=2y = 2.

The curves intersect at y=0y = 0 and y=2y = 2.

Step 2: Set up the volume integral

To compute the volume formed by rotating around the y-axis, we use the method of cylindrical shells. The formula for the volume of a solid formed by rotating about the y-axis is given by: V=2πabxf(y)dyV = 2\pi \int_{a}^{b} x \cdot f(y) \, dy where xx is the radius and f(y)f(y) is the function for the height of the shell.

  • For y3=xy^3 = x, the radius is x=y3x = y^3.
  • For x=4yx = 4y, the radius is x=4yx = 4y.

We will subtract the inner radius (represented by y3y^3) from the outer radius (represented by 4y4y) over the interval y=0y = 0 to y=2y = 2.

The volume is: V=2π02(4yy3)ydyV = 2\pi \int_0^2 (4y - y^3) \, y \, dy This simplifies to: V=2π02(4y2y4)dyV = 2\pi \int_0^2 (4y^2 - y^4) \, dy

Step 3: Integrate

Now, compute the integral: 02(4y2y4)dy=[4y33y55]02\int_0^2 (4y^2 - y^4) \, dy = \left[\frac{4y^3}{3} - \frac{y^5}{5}\right]_0^2 Evaluating at the bounds:

= \left(\frac{4(8)}{3} - \frac{32}{5}\right) = \left(\frac{32}{3} - \frac{32}{5}\right)$$ Now, find a common denominator and subtract: $$\frac{32}{3} - \frac{32}{5} = \frac{160}{15} - \frac{96}{15} = \frac{64}{15}$$ ### Step 4: Multiply by $$2\pi$$ Finally, multiply by $$2\pi$$ to get the volume: $$V = 2\pi \cdot \frac{64}{15} = \frac{128\pi}{15}$$ ### Final Answer: The volume of the solid formed by rotating the region about the y-axis is: $$V = \frac{128\pi}{15}$$ Would you like more details on any part of this process? --- Here are 5 related questions: 1. How would the volume change if the region was rotated about the x-axis instead? 2. Can you compute the surface area of the same solid of revolution? 3. How would you calculate the volume if the curves were $$x = 4y$$ and $$y^2 = x$$? 4. What method could be used to calculate the volume if the curves enclosed a different region? 5. How does the method of disks differ from the method of cylindrical shells? **Tip:** When solving volume problems involving rotation, always visualize the region and identify the axis of rotation carefully before setting up the integral.

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Math Problem Analysis

Mathematical Concepts

Integral Calculus
Volume of Solids of Revolution
Cylindrical Shell Method

Formulas

Volume of a solid of revolution: V = 2π ∫ x f(y) dy
Intersection of curves: Set x = 4y equal to x = y^3

Theorems

Method of Cylindrical Shells
Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12 or Early College (Calculus Level)