Find the volume of the solid formed by rotating the triangular region with vertices at (1,2), (3,2), and (3,4) about the y-axis.
Understand the Problem
The question is asking for the volume of a solid formed by rotating a triangular region around the y-axis. This involves using calculus, specifically the method of cylindrical shells or the disk method, to determine the volume based on the given vertices of the triangle.
Answer
The volume is \( 24\pi \).
Answer for screen readers
The volume of the solid formed by rotating the triangular region about the y-axis is ( V = 24\pi ).
Steps to Solve
- Identify the Triangle Vertices
The vertices of the triangle are:
- ( A(1, 2) )
- ( B(3, 2) )
- ( C(3, 4) )
- Determine the Shape of the Triangle
The base of the triangle lies between points ( A(1, 2) ) and ( B(3, 2) ).
The height of the triangle extends from point ( B(3, 2) ) to point ( C(3, 4) ).
- Set Up the Volume Integral Using the Shell Method
For rotation around the y-axis, we can use the cylindrical shell method. The volume ( V ) is given by: $$ V = 2 \pi \int_{y_1}^{y_2} (radius)(height) , dy $$ In this case, from ( y = 2 ) to ( y = 4 ).
- Define the Radius and Height
-
Radius: The distance from the y-axis to the edge of the triangle (horizontal line) at a height ( y ). The rightmost edge (line ( BC )) is ( x = 3 ), so the radius is ( x = 3 ).
-
Height: The difference in the x-values, which is from point ( A ) at ( x = 1 ) to ( B ) at ( x = 3 ).
The formula for height is:
$$ \text{height} = 3 - 1 = 2 $$
- Write and Evaluate the Integral
We substitute the radius and height: $$ V = 2 \pi \int_{2}^{4} (3)(2) , dy $$ Evaluating the integral: $$ V = 2 \pi \int_{2}^{4} 6 , dy $$ $$ V = 2 \pi \left[ 6y \right]_{2}^{4} $$ $$ V = 2 \pi (6(4) - 6(2)) $$ $$ V = 2 \pi (24 - 12) $$ $$ V = 2 \pi (12) $$ $$ V = 24 \pi $$
The volume of the solid formed by rotating the triangular region about the y-axis is ( V = 24\pi ).
More Information
The method of cylindrical shells is particularly useful for finding volumes of solids of revolution, especially when rotating around axes other than the x-axis. In this case, the triangle forms a prism-like structure when rotated.
Tips
- Forgetting to adjust for the correct limits of integration.
- Mixing up the radius and height for the integral.
- Not squaring the radius in case of disk methods, though the shell method does not require that adjustment.
AI-generated content may contain errors. Please verify critical information
