21D Discussion (Week 4) Robert Bassett UC Davis January 28, 2014 Things to Remember The area of a closed, bounded plane region R is Z Z A= dA. R The average value of f over R is Z Z 1 fdA Area of R R 2 Things to Remember The area of a closed, bounded plane region R is Z Z A= dA. R The average value of f over R is Z Z 1 fdA Area of R R The area differential, dA is polar coordinates is dA = rdrdθ The area of a closed and bounded region R in the polar plane is Z Z A= rdrdθ R 3 Formulas for translating between polar and cartesian coordinates: x = r cos(θ) y = r sin(θ) r2 = x2 + y2 y θ = tan−1 ( ) x 4 More Stuff to Remember The volume of a closed, bounded region D in space is Z Z Z V = dV D The average value of F over D is Z Z Z 1 FdV volume of D D Don’t forget these trig formulas! sin2 (θ) + cos2 (θ) = 1 And the corresponding ones for tan and cot 1 − cos(2θ) sin2 (θ) = 2 1 + cos(2θ) cos2 (θ) = 2 5 Mass and first Moments Formulas Given a density distribution δ Two Dimensions Mass: Z Z M= δdA R First moments: Z Z Z Z My = xδdA, Mx = R Center of Mass: x= y δdA R My Mx , y= M M 6 Three Dimensions Mass: Z Z Z M= δdV D First moments (about planes): Z Z Z Z Z Z Z Z Z y δdV , Mxy = δdV Myz = xδdV , Mxz = D . Center of mass: x= z Myz Mxy Mxz , y= , z= M M M And the centroid is the found by computing the center of mass with δ=1 7 Moments of inertia 2 Dimensions About the x-axis: Z Z y 2 δdA Z Z x 2 δdA Ix = About the y -axis: Iy = About a line L: Z Z IL = r 2 (x, y )δdA where r (x, y ) is the distance from (x, y ) to L. 8 3 Dimensions About the x-axis: Z Z Z (y 2 + z 2 )δdV Z Z Z (x 2 + z 2 )δdV Z Z Z (x 2 + y 2 )δdV Ix = About the y -axis: Iy = About the z-axis: Iz = About a line L: Z Z Z IL = r 2 δdV where r (x, y , z) is the distrance from (x, y , z) to L. 9 Formulas for Cylindrical and Spherical Coordinates Cylindrical: x = r cos(θ), y = r sin(θ), z = z y r 2 = x 2 + y 2 , tan(θ) = x 10 Formulas for Cylindrical and Spherical Coordinates Cylindrical: x = r cos(θ), y = r sin(θ), z = z y r 2 = x 2 + y 2 , tan(θ) = x Spherical: x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), z = ρ cos(φ) p ρ = x2 + y2 + z2 In spherical dV = ρ2 sin(φ)dρdφdθ. Figure: A nice picture explaining spherical coordinates 12 Example Compute the volume of a half-sphere. Set up the integrals in both Cartesian and Spherical Coordinates. 13 Example Show that the centroid of a solid right circular cone is one-fourth of the way from the base to the vertex. (This problem generalizes, but for now just use the equation of the cone x 2 + y 2 = (z − 1)2 ). Appeal to symmetry! 15