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coverband
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1. In Murray Spiegel's Advanced Calculus, in solved problem 10.18 he describes a cylinder as having equation x^2 + y^2 = ax. Surely this equation is for a 2-D object. Whats going on as a cylinder is 3D?
No, it's a 3D object, with z being arbitrary. If you look at the trace of this cylinder in the x-y plane (i.e., z = 0), you get a circle. For any z = k, some constant, you also get a circle, but this time in the plane z = k.coverband said:1. In Murray Spiegel's Advanced Calculus, in solved problem 10.18 he describes a cylinder as having equation x^2 + y^2 = ax. Surely this equation is for a 2-D object. Whats going on as a cylinder is 3D?
The equation of a cylinder is given by x2 + y2 = r2, where r is the radius of the base and x and y are the coordinates of any point on the cylinder's surface.
The variables in the equation of a cylinder are x, y, and r. x and y represent the coordinates of any point on the cylinder's surface, and r represents the radius of the base of the cylinder.
The equation of a cylinder is different from the equation of a circle because it includes two variables, x and y, instead of just one. This is because a cylinder is a three-dimensional shape, while a circle is a two-dimensional shape.
No, the equation of a cylinder cannot be used to directly find the volume or surface area. However, the equation can be used to find the radius of the base, which can then be used in the formula for volume (V = πr2h) and surface area (A = 2πrh + 2πr2).
The equation of a cylinder remains the same for a horizontal or tilted cylinder. The only difference is that the coordinates of the points on the cylinder's surface may change depending on the orientation of the cylinder, but the equation remains x2 + y2 = r2.