I seriously doubt that yo can "substitute d[itex]\theta[/itex]dz" without multiplying by r but I don't understand what coordinates you are using. In converting from Cartesian, x, y, z, coordinates to cylindrical r, [itex]\theta[/itex], z the "differential of volume is r dr d[itex]\theta[/itex] dz. The only kind of integral I can think of that would involve "d[itex]\theta[/itex] would be on the surface of a cylinder. And even then, you would multiply by the radius of the cylinder. That is because lines of constant [itex]\theta[/itex] grow farther apart as r increases.Niles said:Homework Statement
When dealing with an integral integrated with respect to dxdy, I can convert this to polar coordinates, and then integrate with respect to dr d\theta. But I have to multiply with a "r" before integrating.
If I am dealing with an integral with respect to dydz, I can substitute this with d\theta dz, but not have to multiply with "r". Why is that? (And is what I wrote even correct?)
Cartesian coordinates are a system used to locate points in a two-dimensional space. They consist of two perpendicular number lines, known as the x-axis and y-axis, which intersect at the origin (0,0). The x-axis represents horizontal distance and the y-axis represents vertical distance.
Polar coordinates are an alternative system for locating points in a two-dimensional space. They use a distance and angle measurement to describe a point's position. The distance from the origin is known as the radius, and the angle from a reference line, usually the positive x-axis, is known as the polar angle.
To convert from Cartesian coordinates (x,y) to polar coordinates (r,θ), you can use the equations r = √(x²+y²) and θ = tan^-1(y/x). To convert from polar coordinates to Cartesian coordinates, you can use the equations x = rcosθ and y = rsinθ.
In Cartesian coordinates, an integral is a mathematical concept used to find the area under a curve on a graph. It is represented by the symbol ∫ and involves finding the antiderivative of a function. This can also be thought of as finding the total accumulation of a quantity over a given interval.
In polar coordinates, integrals are used to find the area of a region bounded by a polar curve. The integration process involves converting the polar equation into a form that can be integrated using the same techniques as Cartesian integrals. This can also be thought of as finding the total accumulation of a quantity over a given region in a polar coordinate system.