技术问询：求由y=1与y=x²围成区域绕y=2旋转的立体体积

Hey there! Let's work through this volume problem step by step—you've already got the graph drawn, which is half the battle, so great job on that. Here's how to set up and solve the integral:

First, we need to know where the two curves intersect to define our x-range. Set ( y = 1 ) equal to ( y = x^2 ):

x² = 1 → x = -1 and x = 1

So we'll integrate from ( x = -1 ) to ( x = 1 ). Since the region is symmetric across the y-axis, we can also compute the integral from 0 to 1 and double it to simplify calculations.

When rotating around a horizontal line like ( y = 2 ), the washer method is ideal—each vertical slice of our region (at a fixed x) will rotate into a "washer" (a circle with a hole in the middle). We just need to define two radii:

• Outer radius: The distance from the rotation axis ( y=2 ) down to the lower curve ( y=x² ). That's ( 2 - x² ).
• Inner radius: The distance from the rotation axis ( y=2 ) down to the upper curve ( y=1 ). That's ( 2 - 1 = 1 ).

The volume of a washer is ( \pi \times (\text{outer radius}^2 - \text{inner radius}^2) ). Integrate this over our x-range to get the total volume:

V = π ∫₋₁¹ [(2 - x²)² - (1)²] dx

Since the integrand is an even function (all powers of x are even), we can simplify this to:

V = 2π ∫₀¹ [(2 - x²)² - 1] dx

First, expand the squared term in the integrand:

(2 - x²)² = 4 - 4x² + x⁴

Subtract the inner radius squared (1) to get the simplified integrand:

4 - 4x² + x⁴ - 1 = 3 - 4x² + x⁴

Now compute the definite integral:

∫₀¹ (3 - 4x² + x⁴) dx = [3x - (4/3)x³ + (1/5)x⁵]₀¹

Plug in the upper limit (1) and subtract the lower limit (0):

3(1) - (4/3)(1)³ + (1/5)(1)⁵ - 0 = 3 - 4/3 + 1/5

通分计算（分母15）：

3 = 45/15, 4/3 = 20/15, 1/5 = 3/15
45/15 - 20/15 + 3/15 = 28/15

Multiply by ( 2π ) to get the final volume:

V = 2π × (28/15) = 56π/15

Just to make sure this makes sense: the volume should be larger than the cylinder formed by rotating ( y=1 ) from -1 to 1 around ( y=2 ) (which is ( 2π )), and ( 56π/15 ≈ 3.73π ) fits that—since we're including the area below ( y=1 ) down to ( y=x² ), which adds extra volume when rotated.

内容的提问来源于stack exchange，提问作者Sam Goodin