[ V = \int_c^d A(y) , dy ]
Here, (s) is typically the length of the cross‑section at a given (x) or (y), found as the difference between two bounding curves. Problem: The base of a solid is the region bounded by (y = \sqrtx), (y = 0), and (x = 4). Cross‑sections perpendicular to the x‑axis are squares whose bases lie in the base region. Find the volume. volume by cross section practice problems pdf
Base: circle (x^2 + y^2 = 9). Cross sections perpendicular to the x‑axis are equilateral triangles. Find volume. [ V = \int_c^d A(y) , dy ]
| Shape | Area formula | |-------|---------------| | Square (side = (s)) | (A = s^2) | | Equilateral triangle (side = (s)) | (A = \frac\sqrt34 s^2) | | Right isosceles triangle (leg = (s)) | (A = \frac12 s^2) | | Semicircle (diameter = (s)) | (A = \frac\pi8 s^2) | | Rectangle (height = (h), base = (s)) | (A = h \cdot s) | Find the volume
Common cross‑section shapes (when slices are perpendicular to the axis):
For cross sections :