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Section 16.5 Integrals in Cylindrical and Spherical Coordinates

Section 16.5 Integrals in Cylindrical and Spherical Coordinates. In the last section we looked at integrating over a region in the xy -plane given in polar coordinates We can extend polar coordinates into 3 space by adding in the z -axis The result is Cylindrical Coordinates

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Section 16.5 Integrals in Cylindrical and Spherical Coordinates

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  1. Section 16.5Integrals in Cylindrical and Spherical Coordinates

  2. In the last section we looked at integrating over a region in the xy-plane given in polar coordinates • We can extend polar coordinates into 3 space by adding in the z-axis • The result is Cylindrical Coordinates • Just as some double integrals are easier to do in polar coordinates, some triple integrals will be easier to compute in Cylindrical Coordinates

  3. z .(r, θ, z) y r θ .(r, θ, 0) x Cylindrical Coordinates • We can represent points in 3 space with

  4. Cylindrical Coordinates • What type of surfaces do we get if r = c where c is a constant? • What type of surfaces do we get if θ = c where c is a constant? • What type of surfaces do we get if z = c where c is a constant? • These are sometimes referred to as the fundamental surfaces • Regions that are most easily described in cylindrical coordinates are those whose boundaries are fundamental surfaces

  5. Integration in Cylindrical Coordinates • Recall that in polar coordinates we found that dA = rdrdθ • This was based on the fact that ΔA≈ΔrΔθ • Now in rectangular coordinates ΔV≈ΔxΔyΔz andΔA≈ΔxΔy so ΔV≈ΔAΔz • Putting these two lines together we get ΔV≈rΔrΔθΔz • Just as with other iterated integrals, our order of integration will depend on our problem • Let’s take a look at the first 2 problems on the worksheet

  6. z .(r, θ, z) = (x, y, z) ρ y r θ .(r, θ, 0) = (x, y, 0) x Spherical Coordinates • We can represent points in 3 space using

  7. Spherical Coordinates • What type of surfaces do we get if ρ = c where c is a constant? • What type of surfaces do we get if θ = c where c is a constant? • What type of surfaces do we get if = c where c is a constant? • These are sometimes referred to as the fundamental surfaces • Regions that are most easily described in spherical coordinates are those whose boundaries are fundamental surfaces

  8. Integration in Spherical Coordinates • We need to express the volume element, dV, in spherical coordinates • Let’s take a look at what a volume element looks like in spherical coordinates • We can see • When we integrate in spherical coordinates, we have • Let’s revisit the second problem on the worksheet • Now let’s try some of the other problems

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