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Calculus Volume 3

Key Concepts

Calculus Volume 3Key Concepts

Key Concepts

5.1 Double Integrals over Rectangular Regions

  • We can use a double Riemann sum to approximate the volume of a solid bounded above by a function of two variables over a rectangular region. By taking the limit, this becomes a double integral representing the volume of the solid.
  • Properties of double integral are useful to simplify computation and find bounds on their values.
  • We can use Fubini’s theorem to write and evaluate a double integral as an iterated integral.
  • Double integrals are used to calculate the area of a region, the volume under a surface, and the average value of a function of two variables over a rectangular region.

5.2 Double Integrals over General Regions

  • A general bounded region DD on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.
  • To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.
  • We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.
  • We can use Fubini’s theorem for improper integrals to evaluate some types of improper integrals.

5.3 Double Integrals in Polar Coordinates

  • To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.
  • The area dAdA in polar coordinates becomes rdrdθ.rdrdθ.
  • Use x=rcosθ,x=rcosθ, y=rsinθ,y=rsinθ, and dA=rdrdθdA=rdrdθ to convert an integral in rectangular coordinates to an integral in polar coordinates.
  • Use r2=x2+y2r2=x2+y2 and θ=tan−1(yx)θ=tan−1(yx) to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.
  • To find the volume in polar coordinates bounded above by a surface z=f(r,θ)z=f(r,θ) over a region on the xyxy-plane, use a double integral in polar coordinates.

5.4 Triple Integrals

  • To compute a triple integral we use Fubini’s theorem, which states that if f(x,y,z)f(x,y,z) is continuous on a rectangular box B=[a,b]×[c,d]×[e,f],B=[a,b]×[c,d]×[e,f], then
    Bf(x,y,z)dV=efcdabf(x,y,z)dxdydzBf(x,y,z)dV=efcdabf(x,y,z)dxdydz

    and is also equal to any of the other five possible orderings for the iterated triple integral.
  • To compute the volume of a general solid bounded region EE we use the triple integral
    V(E)=E1dV.V(E)=E1dV.
  • Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation.
  • To compute the average value of a function over a general three-dimensional region, we use
    fave=1V(E)Ef(x,y,z)dV.fave=1V(E)Ef(x,y,z)dV.

5.5 Triple Integrals in Cylindrical and Spherical Coordinates

  • To evaluate a triple integral in cylindrical coordinates, use the iterated integral
    θ=αθ=βr=g1(θ)r=g2(θ)z=u1(r,θ)z=u2(r,θ)f(r,θ,z)rdzdrdθ.θ=αθ=βr=g1(θ)r=g2(θ)z=u1(r,θ)z=u2(r,θ)f(r,θ,z)rdzdrdθ.
  • To evaluate a triple integral in spherical coordinates, use the iterated integral
    θ=αθ=βρ=g1(θ)ρ=g2(θ)φ=u1(r,θ)φ=u2(r,θ)f(ρ,θ,φ)ρ2sinφdφdρdθ.θ=αθ=βρ=g1(θ)ρ=g2(θ)φ=u1(r,θ)φ=u2(r,θ)f(ρ,θ,φ)ρ2sinφdφdρdθ.

5.6 Calculating Centers of Mass and Moments of Inertia

Finding the mass, center of mass, moments, and moments of inertia in double integrals:

  • For a lamina RR with a density function ρ(x,y)ρ(x,y) at any point (x,y)(x,y) in the plane, the mass is m=Rρ(x,y)dA.m=Rρ(x,y)dA.
  • The moments about the x-axisx-axis and y-axisy-axis are
    Mx=Ryρ(x,y)dAandMy=Rxρ(x,y)dA.Mx=Ryρ(x,y)dAandMy=Rxρ(x,y)dA.
  • The center of mass is given by x=Mym,y=Mxm.x=Mym,y=Mxm.
  • The center of mass becomes the centroid of the plane when the density is constant.
  • The moments of inertia about the xaxis,xaxis, yaxis,yaxis, and the origin are
    Ix=Ry2ρ(x,y)dA,Iy=Rx2ρ(x,y)dA,andI0=Ix+Iy=R(x2+y2)ρ(x,y)dA.Ix=Ry2ρ(x,y)dA,Iy=Rx2ρ(x,y)dA,andI0=Ix+Iy=R(x2+y2)ρ(x,y)dA.

Finding the mass, center of mass, moments, and moments of inertia in triple integrals:

  • For a solid object QQ with a density function ρ(x,y,z)ρ(x,y,z) at any point (x,y,z)(x,y,z) in space, the mass is m=Qρ(x,y,z)dV.m=Qρ(x,y,z)dV.
  • The moments about the xy-plane,xy-plane, the xz-plane,xz-plane, and the yz-planeyz-plane are
    Mxy=Qzρ(x,y,z)dV,Mxz=Qyρ(x,y,z)dV,Myz=Qxρ(x,y,z)dV.Mxy=Qzρ(x,y,z)dV,Mxz=Qyρ(x,y,z)dV,Myz=Qxρ(x,y,z)dV.
  • The center of mass is given by x=Myzm,y=Mxzm,z=Mxym.x=Myzm,y=Mxzm,z=Mxym.
  • The center of mass becomes the centroid of the solid when the density is constant.
  • The moments of inertia about the yz-plane,yz-plane, the xz-plane,xz-plane, and the xy-planexy-plane are
    Ix=Q(y2+z2)ρ(x,y,z)dV,Iy=Q(x2+z2)ρ(x,y,z)dV,Iz=Q(x2+y2)ρ(x,y,z)dV.Ix=Q(y2+z2)ρ(x,y,z)dV,Iy=Q(x2+z2)ρ(x,y,z)dV,Iz=Q(x2+y2)ρ(x,y,z)dV.

5.7 Change of Variables in Multiple Integrals

  • A transformation TT is a function that transforms a region GG in one plane (space) into a region RR in another plane (space) by a change of variables.
  • A transformation T:GRT:GR defined as T(u,v)=(x,y)T(u,v)=(x,y) (orT(u,v,w)=(x,y,z))(orT(u,v,w)=(x,y,z)) is said to be a one-to-one transformation if no two points map to the same image point.
  • If ff is continuous on R,R, then Rf(x,y)dA=Sf(g(u,v),h(u,v))|(x,y)(u,v)|dudv.Rf(x,y)dA=Sf(g(u,v),h(u,v))|(x,y)(u,v)|dudv.
  • If FF is continuous on R,R, then
    RF(x,y,z)dV=GF(g(u,v,w),h(u,v,w),k(u,v,w))|(x,y,z)(u,v,w)|dudvdw=GH(u,v,w)|J(u,v,w)|dudvdw.RF(x,y,z)dV=GF(g(u,v,w),h(u,v,w),k(u,v,w))|(x,y,z)(u,v,w)|dudvdw=GH(u,v,w)|J(u,v,w)|dudvdw.
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