Express 5.5 ℎ−1 in −1.
Answers
Step-by-step explanation:
integrals, but here we need to distinguish between cylindrical symmetry and spherical symmetry. In this section we convert triple integrals in rectangular coordinates into a triple integral in either cylindrical or spherical coordinates.
Also recall the chapter opener, which showed the opera house l’Hemisphèric in Valencia, Spain. It has four sections with one of the sections being a theater in a five-story-high sphere (ball) under an oval roof as long as a football field. Inside is an IMAX screen that changes the sphere into a planetarium with a sky full of 9000 twinkling stars. Using triple integrals in spherical coordinates, we can find the volumes of different geometric shapes like these.
Review of Cylindrical Coordinates
As we have seen earlier, in two-dimensional space ℝ2, a point with rectangular coordinates (x,y) can be identified with (r,θ) in polar coordinates and vice versa, where x=rcosθ, y=rsinθ, r2=x2+y2 and tanθ=(
y
x
) are the relationships between the variables.
In three-dimensional space ℝ3, a point with rectangular coordinates (x,y,z) can be identified with cylindrical coordinates (r,θ,z) and vice versa. We can use these same conversion relationships, adding z as the vertical distance to the point from the xy-plane as shown in the following figure.
In xyz space, a point is shown (x, y, z). There is also a depiction of it in polar coordinates as (r, theta, z).
Answer:
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