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 Just as some double integrals are easier to do in polar coordinates, some triple integrals will be easier to compute in Cylindrical Coordinates

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

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

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 and ΔA ≈ 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

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

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

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