我没懂为什么底面是圆，横截面却是正方形，给我讲一下---**Question 1** **Question Stem:** 某个立体的底面是半径为 r 的圆。所有垂直于底面一条直径的横截面都是正方形。求该立体的体积。 **Other Relevant Text:** The problem describes a three-dimensional solid. - Its base is a circle with radius 'r'. - All cross-sections perpendicular to one diameter of the base are squares. - The question asks for the volume of this solid.

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Let's understand this geometric problem step by step. We have a solid with a circular base of radius r. The key insight is that cross-sections perpendicular to a diameter are squares. This might seem confusing at first - how can a circular base produce square cross-sections? The answer lies in how we slice the solid. When we cut perpendicular to a diameter, we get a chord of the circle, and this chord length determines the side length of the square cross-section. Now let's analyze how the cross-sections work mathematically. At any position x along the diameter, we can find the chord length using the Pythagorean theorem. The chord length equals 2 times the square root of r squared minus x squared. This chord length becomes the side length of our square cross-section. As x varies from negative r to positive r, the chord length changes, creating squares of different sizes. At the center where x equals zero, we get the maximum square with side length 2r. At the edges where x approaches plus or minus r, the square shrinks to zero size. Now we set up the integration to find the volume. The area of each square cross-section at position x is the side length squared. Since the side length is 2 times the square root of r squared minus x squared, the area becomes 4 times r squared minus x squared. To find the total volume, we integrate this area function from negative r to positive r. This gives us the integral of 4 times r squared minus x squared with respect to x, from negative r to r. The graph shows how the cross-sectional area varies along the diameter, and the shaded region represents the total volume we need to calculate. Now let's evaluate the integral step by step. We start with the integral of 4 times r squared minus x squared from negative r to r. First, we factor out the constant 4. Then we find the antiderivative: r squared x minus x cubed over 3. Next, we evaluate this at the limits. At x equals r, we get r cubed minus r cubed over 3. At x equals negative r, we get negative r cubed plus r cubed over 3. Subtracting these gives us 2 r cubed minus 2 r cubed over 3. Simplifying this fraction, we get 4 r cubed over 3. Multiplying by 4 gives us our final answer: 16 r cubed over 3. This is the volume of our solid with circular base and square cross-sections. 我们来理解这个几何问题。立体的底面是一个半径为r的圆。关键是理解什么是垂直于底面直径的横截面。我们先选定底面圆的一条直径作为参考。 横截面是指用垂直于直径的平面来切割立体时产生的截面。在圆的边缘附近，弦长较短，对应的正方形横截面较小；在圆心附近，弦长最大，正方形横截面也最大。这就是为什么底面是圆，横截面却是正方形的原因。 为了计算体积，我们建立坐标系。将圆心放在原点，直径沿x轴方向。对于x轴上任意位置x，垂直切面与圆的交线是一条弦，弦长为2倍根号下r平方减x平方。既然横截面是正方形，正方形的边长就等于这个弦长。 现在我们计算体积。在位置x处，正方形横截面的面积是边长的平方，即4倍括号r平方减x平方。要求整个立体的体积，我们需要对这个面积函数从负r到正r进行积分。这个积分的几何意义就是将所有横截面叠加起来。 现在进行最终计算。对4倍r平方减x平方从负r到r积分，首先求出原函数r平方x减三分之x立方。代入上下限计算得到4倍括号2r立方减三分之2r立方，化简后得到4倍三分之4r立方，最终结果是三分之16r立方。这就是该立体的体积公式。