Final answer:
The volume of a solid with a semicircular base and square cross-sections is calculated by integrating the area of the squares across the base of the semicircle, which is derived from the equation of a semicircle and the area of a square.
Step-by-step explanation:
To find the volume of a solid with a semicircular base where cross-sections perpendicular to the x-axis are squares, we can use calculus, specifically the method of slicing. The volume of the solid can be found by integrating the area of the square cross-sections along the base of the semicircle.
Let the semicircle have radius r. The equation of the semicircle is y = √(r² - x²). Since the cross-sections are squares, the side length of each square is 2y. Therefore, the area of each square is A = (2y)² = 4(r² - x²).
The volume V is then the integral of A with respect to x, from -r to r, giving us V = ∫-rr 4(r² - x²) dx.
Using the formula for the volume of a cylinder V = Ah as a reference, we realize the volume is essentially the area of the semicircle times the height, where in this case the height is the thickness of a very thin slice along the axis.