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.

Steps to Solve

1. Identify the Vertices of the Triangle

The vertices of the triangular region are given as ( (1, 2) ), ( (3, 2) ), and ( (3, 4) ).

2. Find the Area of the Triangle

Using the formula for the area of a triangle given vertices, the area ( A ) can be computed using:

$$ A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| $$

Plugging in the vertices:

$$ A = \frac{1}{2} \left| 1(2-4) + 3(4-2) + 3(2-2) \right| $$

Calculating gives:

$$ A = \frac{1}{2} \left| 1(-2) + 3(2) + 3(0) \right| = \frac{1}{2} \left| -2 + 6 \right| = \frac{1}{2} \cdot 4 = 2 $$

3. Set Up the Volume Integral

To find the volume ( V ) of the solid obtained by rotating the triangle about the y-axis, we will use the washer method. The height of the triangle can be described with the equation of the lines. Between ( y = 2 ) and ( y = 4 ) the right endpoint at ( x = 3 ) and left endpoint at ( x = 1 ) gives us:

$$ V = \pi \int_{2}^{4} [(3)^2 - (1)^2] , dy $$

4. Compute the Integral

Calculate the integral:

$$ V = \pi \int_{2}^{4} [9 - 1] , dy $$

This simplifies to:

$$ V = \pi \int_{2}^{4} 8 , dy $$

Evaluating the integral gives:

$$ V = \pi [8y]_{2}^{4} = \pi [8(4) - 8(2)] = \pi [32 - 16] = 16\pi $$