Find the volume of the solid generated by revolving the region enclosed by x = √(5)y² and the lines y = 1, x = 0, y = -1 about the y-axis.
Understand the Problem
The question is asking for the volume of a solid formed by revolving a specific region around the y-axis. This involves using calculus, specifically methods for finding volumes of solids of revolution.
Answer
$V = 2\pi$
Answer for screen readers
The volume of the solid generated is $V = 2\pi$.
Steps to Solve
- Identify the region enclosed by the curves
The region is bounded by the curve $x = \sqrt{5} y^2$, the lines $y = 1$, $y = -1$, and $x = 0$.
- Set up the volume integral
To find the volume generated by rotating the region around the y-axis, we will use the formula for the volume of revolution: $$ V = \pi \int_{a}^{b} [f(y)]^2 , dy $$ Here, $f(y) = \sqrt{5}y^2$, the limits of integration $a = -1$ and $b = 1$.
- Evaluate the integral
Substituting $f(y)$ into the volume formula: $$ V = \pi \int_{-1}^{1} (\sqrt{5} y^2)^2 , dy $$ This simplifies to: $$ V = \pi \int_{-1}^{1} 5 y^4 , dy $$
- Calculate the definite integral
Computing the integral: $$ \int 5 y^4 , dy = 5 \cdot \frac{y^5}{5} = y^5 $$ Thus, $$ V = \pi \left[ y^5 \right]_{-1}^{1} $$ Calculating this gives: $$ V = \pi \left( 1^5 - (-1)^5 \right) = \pi (1 - (-1)) = \pi (1 + 1) = 2\pi $$
The volume of the solid generated is $V = 2\pi$.
More Information
This result shows the volume of the solid of revolution created by rotating the specified region about the y-axis. Volumes of solids of revolution are often computed using integrals, specifically when curves and boundaries are involved.
Tips
- Forgetting absolute values: When calculating the definite integral, ensure that you account for the signs correctly when integrating odd functions over symmetric intervals.
- Limits of integration: Make sure to correctly identify the limits of integration based on the boundaries given in the problem.
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