Question

Using cylindrical coordinates, find the volume above the xy plane and the cylinder 2 + 2 = 25 and below the plane + − 2 = 1.

Answer 1

To convert from rectangular to cylindrical coordinates, we use the conversion x=r\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta and y=r\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta . To convert from cylindrical to rectangular coordinates, we use {r}^{2}={x}^{2}+{y}^{2} and \theta ={\text{tan}}^{-1}\left(\frac{y}{x}\right). The z-coordinate remains the same in both cases.

In the two-dimensional plane with a rectangular coordinate system, when we say x=k (constant) we mean an unbounded vertical line parallel to the y-axis and when y=l (constant) we mean an unbounded horizontal line parallel to the x-axis. With the polar coordinate system, when we say r=c (constant), we mean a circle of radius c units and when \theta =\alpha (constant) we mean an infinite ray making an angle \alpha with the positive x-axis.

Similarly, in three-dimensional space with rectangular coordinates \left(x,y,z\right), the equations x=k,y=l, and z=m, where k,l, and m are constants, represent unbounded planes parallel to the yz-plane, xz-plane and xy-plane, respectively. With cylindrical coordinates \left(r,\theta ,z\right), by r=c,\theta =\alpha , and z=m, where c,\alpha , and m are constants, we mean an unbounded vertical cylinder with the z-axis as its radial axis; a plane making a constant angle \alpha with the xy-plane; and an unbounded horizontal plane parallel to the xy-plane, respectively. This means that the circular cylinder {x}^{2}+{y}^{2}={c}^{2} in rectangular coordinates can be represented simply as r=c in cylindrical coordinates. (Refer to Cylindrical and Spherical Coordinates for more review.)

Answer 2

Answer:

Hiii

Explanation:

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