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 and , where , 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 equal to : Dividing both sides by (since excludes ): Thus, .
The curves intersect at and .
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: where is the radius and is the function for the height of the shell.
- For , the radius is .
- For , the radius is .
We will subtract the inner radius (represented by ) from the outer radius (represented by ) over the interval to .
The volume is: This simplifies to:
Step 3: Integrate
Now, compute the integral: 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.Ask a new question for Free
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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)
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