Integration using Transformations in Polar, Cylindrical, and Spherical Coordinates

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1 ections 15.4 Integration using Transformations in Polar, Cylindrical, and pherical Coordinates Cylindrical Coordinates pherical Coordinates MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 1 / 15

2 Double Integration with Change of Variables Let T (u, v) = (x(u, v), y(u, v)) be a transformation where T () =, then f (x, y) da xy = f (x(u, v), y(u, v)) (x, y) (u, v) da uv In section 15.6, we verified that G(r, θ) = (r cos(θ), r sin(θ)) is a transformation from polar coordinates to rectangular coordinates. Further, (x, y) (r, θ) = cos(θ) r sin(θ) sin(θ) r cos(θ) = r cos2 (θ) + r sin 2 (θ) = r Therefore, we can integrate over regions better described using polar coordinates in polar coordinates using the transformation G: f (x, y) da xy = f (r cos(θ), r sin(θ)) r da rθ MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 2 / 15

3 Example 1: Evaluate the integral where is the region bounded by the curve x 2 + y 2 = 4. e x2 +y 2 da olution: The region is a circle and can be described in polar coordinates as {(r, θ) r 1 θ < 2π} ince r 2 = x 2 + y 2, e x2 +y 2 da = 2π 1 e r 2 r dr dθ = 2π [ ] e r dθ = 2π 1 (e 1) dθ = π(e 1) 2 MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 3 / 15

4 Triple Integration with Change of Variables Let T (u, v, w) = (x(u, v, w), y(u, v, w), z(u, v, w)) be a transformation where T () =, then f (x, y, z) dv xyz equals f (x(u, v, w), y(u, v, w), z(u, v, w)) (x, y, z) (u, v, w) dv uvw G(r, θ, z) = (r cos(θ), r sin(θ), z) is a transformation from cylindrical coordinates to rectangular coordinates. cos(θ) r sin(θ) (x, y, z) (r, θ, z) = sin(θ) r cos(θ) = r 1 Therefore, we can integrate over solids better described using cylindrical coordinates in cylindrical coordinates using the transformation G: f (x, y, z) dv xyz = f (r cos(θ), r sin(θ), z) r dv rθz MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 4 / 15

5 Example 2: Integrate f (x, y, z) = x 2 + y 2 over the solid contained inside the cylinder x 2 + y 2 = 1, under the plane z = 4, and above the elliptic paraboloid z = 1 x 2 y 2. olution: In cylindrical coordinates, the solid has bounds θ 2π, r 1, and 1 r 2 z 4. x 2 + y 2 dv = 2π r 2 r 3 dz dr dθ = 2π 1 3r 3 + r 5 drdθ = 2π [ 3r r 6 ] 1 dθ 6 = 11π 6 MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 5 / 15

6 pherical Coordinates G(ρ, φ, θ) = (ρ sin(φ) cos(θ), ρ sin(φ) sin(θ), ρ cos(φ)) is a transformation from spherical coordinates to rectangular coordinates. (x, y, z) (ρ, φ, θ) = sin(φ) cos(θ) ρ cos(φ) cos(θ) ρ sin(φ) sin(θ) sin(φ) sin(θ) ρ cos(φ) sin(θ) ρ sin(φ) cos(θ) cos(φ) ρ sin(φ) = ρ 2 sin(φ) Therefore, we can integrate over solids better described using spherical coordinates in spherical coordinates using the transformation G: f (x, y, z) dv xyz = f (ρ, φ, θ) ρ 2 sin(φ) dv ρφθ MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 6 / 15

7 Example 3: Find the volume of the solid contained above the cone z = x 2 + y 2 and below the sphere x 2 + y 2 + z 2 = z. olution: In spherical coordinates, the solid has bounds θ 2π, φ π 4, and ρ cos(φ). 1 dv = = 2π π 4 cos(φ) ρ 2 sin(φ) dρ dφ dθ ( ) π 2π 4 sin(φ) cos 3 (φ) dφ 3 = π 8 MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 7 / 15

8 ection 15.5 Applications of Multiple Integrals (I) Density and Mass (II) Center of Mass MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 8 / 15

9 Applications of Multiple Integrals In MATH 126, we used integrals to represent quantities that could be described as the total amount of something. This involved a two-step procedure for computing such quantities: (1) Approximate the quantity by a sum of N terms. (2) Pass the approximation to the limit as N. (i) Area (ii) Volume (iii) Work (iv) Average (v) Force (vi) Probability We have been calculating area and volumes in sections In chapter 16 we will revisit work. In 15.5, we calculate mass and the center of mass of regions in 2 and 3. MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 9 / 15

10 Density and Mass uppose a lamina occupies a region D of the xy-plane and its density (in units of mass per unit area) at a point (x, y) in D is given by ρ(x, y), where ρ is a continuous function on D. Mass in 2 : lim (n,m) (, ) n m i=1 j=1 ρ(xij, yij ) A = ρ(x, y) da D MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 1 / 15

11 Density and Mass Mass in 3 : ρ(x, y) dv Example 1: Find the mass of the rectangle = [, 2] [, 3] with density ρ(x, y) = xy 2 kg m. olution: ρ(x, y) da = = 3 2 ( 2 xy 2 dx dy ) ( 3 ) x dx y 2 dy = (2)(3) = 6 kg MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 11 / 15

12 Moments and Center of Mass The moment about an axis is the measurement of the tendency a region has to rotate about that axis. The coordinates (x, y) of the center of mass of a lamina occupying the region D and having density function ρ(x, y) and mass m are M x = yρ(x, y) da M y = xρ(x, y) da D x = M y m = 1 xρ(x, y) da m D y = M x m = 1 yρ(x, y) da m D Think of the coordinates as weighted averages. They are the averages of x and y in which the factor ρ assigns more weight to points with larger mass density. MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 12 / 15 D

13 Example 2: Find the mass and center of mass of a triangular lamina with vertices (, ), (1, ), and (, 2) and density function ρ(x, y) = 1 + 3x + y. olution: The lamina has boundaries y = 2 2x, y =, and x =. The mass of the lamina is 1 2 2x m = ρ(x, y) da = (1 + 3x + y) dy dx = 8 3 The center of mass coordinates: D 1 2 2x x = 1 m x(1 + 3x + y) dy dx = 3 8 y = 1 m 1 2 2x y(1 + 3x + y) dy dx = MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 13 / 15

14 Moments and Center of Mass In 3, the moments of a solid are defined not with respect to the axes as in 2, but with respect to the coordinate planes: M yz = xρ(x, y, z) dv x = M yz m M xz = yρ(x, y, z) dv y = M xz m M xy = zρ(x, y, z) dv z = M xy m MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 14 / 15

15 Example 3: Find the center of mass of the tetrahedron of uniform density bounded by the coordinate planes and the plane x + y + z = 1. olution: Let ρ be the density of. The mass of is ρ dv = M yz = M xz = M xy = 1 1 x 1 x y xρ dv = ρ 24 yρ dv = ρ 24 zρ dv = ρ 24 ρ dz dy dx = ρ 6 x = 1 4 y = 1 4 z = 1 4 MATH 127 (ection 15.5) Applications of Multiple Integrals The University of Kansas 15 / 15