Question 1181748
Here's how to find the area of the sections:

**1. Visualize the Problem:** Imagine a sphere and a cylinder intersecting. The cylinder's diameter is equal to the sphere's radius, and one of the cylinder's sides (an element) is aligned with a diameter of the sphere.  The common solid is the region where both the sphere and cylinder exist.

**2. Define Coordinate System:** It's helpful to use a coordinate system. Let the axis of the cylinder and the diameter of the sphere be along the z-axis.  The center of the sphere is at the origin (0,0,0).

**a) Plane Containing the Axis and the Diameter:**

*   This plane is essentially a cross-section through the center of both the sphere and the cylinder.
*   The intersection with the sphere is a circle of radius R.
*   The intersection with the cylinder is a rectangle with height 2R (the diameter of the sphere) and width R (the diameter of the cylinder).
*   The section of the common solid is the *smaller* of these two shapes, which is the rectangle.
*   Area of the rectangle = height * width = 2R * R = 2R²

**b) Plane Perpendicular to the Axis at Midpoint:**

*   This plane is horizontal (parallel to the x-y plane) and passes through the center of the sphere (z = 0).
*   The intersection with the sphere is a circle of radius R.
*   The intersection with the cylinder is a circle of radius R/2.
*   The section of the common solid is the smaller circle, which is the intersection with the cylinder.
*   Area of the circle = π * (radius)² = π * (R/2)² = (πR²)/4

**c) Plane Containing the Axis and Perpendicular to the Plane in (a):**

*   This plane is perpendicular to the plane in (a).  It's effectively a vertical plane passing through the center of the cylinder.
*   The intersection with the sphere is a circle of radius R.
*   The intersection with the cylinder is a rectangle with height 2R and width R.
*   The section of the common solid is again the smaller of the two, which will be two rectangles, each with height 2R and width R/2.
*   Total area of the two rectangles = 2 * (height * width) = 2 * (2R * R/2) = 2R²

**Summary of Areas:**

*   (a) Plane containing the axis and the diameter: 2R²
*   (b) Plane perpendicular to the axis at midpoint: (πR²)/4
*   (c) Plane containing the axis and perpendicular to the plane in (a): 2R²