Questions: What is the volume of the solid of revolution obtained by rotating the region bounded by x=1, x=2, y=0 and y=x^2 around the y-axis

Transcript text: What is the volume of the solid of revolution ottained by rotating the region bounded by $x=1, x=2, y=0$ and $y=x^{2}$ around the y-axis

Solution

Solution Steps

To find the volume of the solid of revolution obtained by rotating the region bounded by $ x=1 $, $ x=2 $, $ y=0 $, and $ y=x^2 $ around the y-axis, we can use the method of cylindrical shells. The volume $ V $ is given by the integral:

\[ V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \]

where $ f(x) = x^2 $, $ a = 1 $, and $ b = 2 $.

Step 1: Define the Region and Method

We need to find the volume of the solid of revolution obtained by rotating the region bounded by $ x=1 $, $ x=2 $, $ y=0 $, and $ y=x^2 $ around the y-axis. We will use the method of cylindrical shells.

Step 2: Set Up the Integral

The volume $ V $ is given by the integral: \[ V = 2\pi \int_{1}^{2} x \cdot x^2 \, dx \] \[ V = 2\pi \int_{1}^{2} x^3 \, dx \]

Step 3: Evaluate the Integral

Evaluate the integral: \[ \int_{1}^{2} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{1}^{2} = \frac{2^4}{4} - \frac{1^4}{4} = \frac{16}{4} - \frac{1}{4} = 4 - 0.25 = 3.75 \]

Step 4: Multiply by $ 2\pi $

Multiply the result by $ 2\pi $: \[ V = 2\pi \cdot 3.75 = \frac{15\pi}{2} \]

Step 5: Convert to Decimal Form

Convert the volume to decimal form: \[ V \approx 23.5619 \]