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Properties and Applications of Double Integrals

  • Functions of two variables: \(f\left( {x,y} \right)\), \(f\left( {u,v} \right)\), \(g\left( {x,y} \right)\)
    Independent variables: \(x\), \(y\), \(u\), \(v\)
    Small changes: \(\Delta {x_i}\), \(\Delta {y_i}\)
    Regions of integration: \(R\), \(S\)
    Real numbers: \(a\), \(b\), \(c\), \(d,\) \(\alpha,\) \(\beta\)
    Polar coordinates: \(r\), \(\theta\)
    Area of a region: \(A\)
    Surface area: \(S\)
    Volume of a solid: \(V\)
    Mass of a lamina: \(m\)
    Density of a lamina: \(\rho\left( {x,y} \right)\)
    First moments: \({M_x}\), \({M_y}\)
    Moments of inertia: \({I_x}\), \({I_y}\), \({I_0}\)
    Charge of a plate: \(Q\)
    Charge density: \(\sigma\left( {x,y} \right)\)
    Coordinates of the center of mass: \(\bar x\), \(\bar y\)
    Average of a function: \(\mu\)
    1. The double integral of a function \(f\left( {x,y} \right)\) over a rectangular region \(\left[ {a,b} \right] \times \left[ {c,d} \right]\) is defined as the limit of the integral sum (Riemann sum):
      \(\require{AMSmath.js}{\large\iint\limits_{\left[ {a,b} \right] \times \left[ {c,d} \right]}\normalsize} {f\left( {x,y} \right)dA} =\) \(\lim\limits_{\substack{ \text{max}\,\Delta {x_i} \to 0\\ \text{max}\,\Delta {y_j} \to 0}} \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {f\left( {{u_i},{v_j}} \right)\Delta {x_i}\Delta {y_j}} },\)
      where \({\left( {{u_i},{v_j}} \right)}\) is some point in the rectangle \(\left( {{x_{i – 1}},{x_i}} \right) \times \left( {{y_{j – 1}},{y_j}} \right)\) and \(\Delta {x_i} = {x_i} – {x_{i – 1}},\) \(\Delta {y_j} = {y_j} – {y_{j – 1}}.\)
    2. The double integral over a rectangular region
    3. The double integral of a function \(f\left( {x,y} \right)\) over a general region \(R\) is defined to be
      \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =\) \({\large\iint\limits_{\left[ {a,b} \right] \times \left[ {c,d} \right]}\normalsize} {g\left( {x,y} \right)dA},\)
      where the rectangle \(\left[ {a,b} \right] \times \left[ {c,d} \right]\) contains the region \(R\). The function \(g\left( {x,y} \right)\) is defined as follows:
      \(g\left( {x,y} \right) = f\left( {x,y} \right)\) when \(f\left( {x,y} \right)\) is in \(R\) and \(g\left( {x,y} \right) = 0\) otherwise.
    4. The double integral over a general region
    5. The double integral of the sum of two functions is equal to the sum of the integrals of these functions
      \({\large\iint\limits_R\normalsize} {\left[ {f\left( {x,y} \right) + g\left( {x,y} \right)} \right]dA} =\) \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} \) \(+\;{\large\iint\limits_R\normalsize} {g\left( {x,y} \right)dA} \)
    6. The double integral of the difference of two functions is equal to the difference of the integrals of these functions:
      \({\large\iint\limits_R\normalsize} {\left[ {f\left( {x,y} \right) – g\left( {x,y} \right)} \right]dA} =\) \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} \) \(-\;{\large\iint\limits_R\normalsize} {g\left( {x,y} \right)dA} \)
    7. A constant factor can be moved across the double integral sign:
      \({\large\iint\limits_R\normalsize} {kf\left( {x,y} \right)dA} =\) \(k{\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} \)
    8. If \(f\left( {x,y} \right) \le g\left( {x,y} \right)\) in a region \(R\), then
      \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} \le\) \({\large\iint\limits_R\normalsize} {g\left( {x,y} \right)dA} \)
    9. If \(f\left( {x,y} \right) \ge 0\) in a region \(R\) and \(S \subset R\), then
      \({\large\iint\limits_S\normalsize} {f\left( {x,y} \right)dA} \le\) \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} \)
    10. Region S in R
    11. If \(f\left( {x,y} \right) \ge 0\) in a region \(R\) and \(R\) and \(S\) are non-overlapping regions, then
      \({\large\iint\limits_{R \cup S}\normalsize} {f\left( {x,y} \right)dA} =\) \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} \) \(+\;{\large\iint\limits_S\normalsize} {f\left( {x,y} \right)dA}.\)
      Here \({R \cup S}\) is the union of the regions of integration \(R\) and \(S.\)
    12. Union of the regions of integration R and S
    13. Iterated integrals and Fubini’s theorem for a region of type \(I\)
      \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =\) \({\large\int\limits_a^b\normalsize} {{\large\int\limits_{p\left( x \right)}^{q\left( x \right)}\normalsize} {f\left( {x,y} \right)dy\,dx} }, \)
      where the region of integration \(R\) is defined by the inequalities
      \(R = \big\{ \left( {x,y} \right) \mid a \le x \le b,\) \(p \left( x \right) \le y \le q\left( x \right)\big\}.\)
    14. Iterated integrals and Fubini’s theorem for a region of type I
    15. Iterated integrals and Fubini’s theorem for a region of type \(II\)
      \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =\) \({\large\int\limits_c^d\normalsize} {{\large\int\limits_{u\left( y \right)}^{v\left( y \right)}\normalsize} {f\left( {x,y} \right)dx\,dy} }, \)
      where the region of integration \(R\) is defined by the inequalities
      \(R = \big\{ \left( {x,y} \right) \mid u\left( y \right) \le x \le v\left( y \right),\) \(c \le y \le d \big\}.\)
    16. Iterated integrals and Fubini’s theorem for a region of type II
    17. Double integrals over rectangular regions
      If \(R\) is a rectangular region \(\left[ {a,b} \right] \times \left[ {c,d} \right]\), then
      \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =\) \({\large\int\limits_a^b\normalsize} {\left( {{\large\int\limits_c^d\normalsize} {f\left( {x,y} \right)dy} } \right)dx} =\) \({\large\int\limits_c^d\normalsize} {\left( {\large\int\limits_a^b\normalsize {f\left( {x,y} \right)dx} } \right)dy} \)
      In the special case when the integrand \(f\left( {x,y} \right)\) can be written as the product \(g\left( x \right) h\left( y \right)\), the double integral is given by
      \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dxdy} =\) \({\large\iint\limits_R\normalsize} {g\left( x \right)h\left( y \right)dxdy} =\) \(\left( {{\large\int\limits_a^b\normalsize} {g\left( x \right)dx} } \right)\left( {{\large\int\limits_c^d\normalsize} {h\left( y \right)dy} } \right)\)
    18. Change of variables
      \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dxdy} =\) \({\large\iint\limits_S\normalsize} {f\left[ {x\left( {u,v} \right),y\left( {u,v} \right)} \right] }\) \({ \left| {\large\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {u,v} \right)}}\normalsize} \right|dudv}, \)
      where \(\left|{\large\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {u,v} \right)}}\normalsize}\right| =\) \( \left| {\begin{array}{*{20}{c}} {\large\frac{{\partial x}}{{\partial u}}\normalsize} & {\large\frac{{\partial x}}{{\partial v}}\normalsize}\\ {\large\frac{{\partial y}}{{\partial u}}\normalsize} & {\large\frac{{\partial y}}{{\partial v}}\normalsize} \end{array}} \right|\) \( \ne 0\) is the jacobian of the transformation \(\left( {x,y} \right) \to \left( {u,v} \right)\) and \(S\) is the pullback of the region \(R\) which can be computed by substituting \(x = x\left( {u,v} \right)\), \(y = y\left( {u,v} \right)\) into the definition of \(R\).
    19. Polar coordinates
      \(x = r\cos \theta,\) \(y = r\sin \theta \)
    20. Polar coordinates
    21. Double integrals in polar coordinates
      The differential \(dxdy\) in polar coordinates is given by the expression
      \(dxdy = \left| {\large\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {r,\theta } \right)}}\normalsize} \right|drd\theta =\) \( rdrd\theta \)
      Let the region of integration \(R\) be defined as follows:
      \(0 \le g\left( \theta \right) \le r \le h\left( \theta \right)\), \(\alpha \le \theta \le \beta, \)
      where \(\beta – \alpha \le 2\pi \). Then
      \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dxdy} =\) \({\large\int\limits_\alpha ^\beta\normalsize} {{\large\int\limits_{g\left( \theta \right)}^{h\left( \theta \right)}\normalsize} {f\left( {r\cos \theta ,r\sin \theta } \right)rdrd\theta } } \)
    22. Double integrals in polar coordinates
    23. Double integral over a polar rectangle
      If the region of integration \(R\) is a polar rectangle given by the inequalities \(0 \le a \le r \le b\), \(\alpha \le \theta \le \beta,\) where \(\beta – \alpha \le 2\pi \), then
      \({\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dxdy} =\) \({\large\int\limits_\alpha ^\beta\normalsize} {{\large\int\limits_a^b\normalsize} {f\left( {r\cos \theta ,r\sin \theta } \right)rdrd\theta } } \)
    24. Double integral over a polar rectangle
    25. Area of a type \(I\) region
      \(A = {\large\int\limits_a^b\normalsize} {{\large\int\limits_{g\left( x \right)}^{h\left( x \right)}\normalsize} {dydx} } \)
    26. Area of a type I region
    27. Area of a type \(II\) region
      \(A = {\large\int\limits_c^d\normalsize} {{\large\int\limits_{p\left( y \right)}^{q\left( y \right)}\normalsize} {dxdy} } \)
    28. Area of a type II region
    29. Volume of a solid
      \(V = {\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} \)
    30. Volume of a solid
      If \(R\) is a type \(I\) region bounded by \(x = a,\) \(x = b,\) \(y = h\left( x \right),\) \(y = g\left( x \right),\) then
      \(V = {\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =\) \( {\large\int\limits_a^b\normalsize} {{\large\int\limits_{h\left( x \right)}^{g\left( x \right)}\normalsize} {f\left( {x,y} \right)dydx} } \)
      If \(R\) is a type \(II\) region bounded by \(y = c,\) \(y = d,\) \(x = q\left( y \right),\) \(x = p\left( y \right),\) then
      \(V = {\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =\) \({\large\int\limits_c^d\normalsize} {{\large\int\limits_{p\left( y \right)}^{q\left( y \right)}\normalsize} {f\left( {x,y} \right)dxdy} } \)
    31. Volume of a solid between two surfaces
      If \(f\left( {x,y} \right) \ge g\left( {x,y} \right)\) over a region \(R\), then the volume of the solid between the surfaces \({z_1}\left( {x,y} \right)\) and \({z_2}\left( {x,y} \right)\) over \(R\) is given by
      \(V = {\large\iint\limits_R\normalsize} {\big[ {f\left( {x,y} \right) }}\) \(-\;{{ g\left( {x,y} \right)} \big]dA} \)
    32. Area and volume in polar coordinates
      Let a region \(S\) be given in polar coordinates in the \(xy\)-plane and bounded by the lines \(\theta = \alpha,\) \(\theta = \beta,\) \(r = h\left( \theta \right),\) \(r = g\left( \theta \right).\) Let also a function \(f\left( {r,\theta} \right)\) be given in the region \(S\). Then the area of the region \(S\) and volume of the solid bounded by the surface \(f\left( {r,\theta} \right)\) are determined by the formulas
      \(A = {\large\iint\limits_S\normalsize} {dA} =\) \({\large\int\limits_\alpha ^\beta\normalsize} {{\large\int\limits_{h\left( \theta \right)}^{g\left( \theta \right)}\normalsize} {rdrd\theta } } ,\) \(V = {\large\iint\limits_S\normalsize} {f\left( {r,\theta } \right)rdrd\theta } \)
    33. Double integral over a polar rectangle
    34. Surface area
      \(S =\) \({\large\iint\limits_R\normalsize} {\sqrt {1 + {{\left( {\large\frac{{\partial z}}{{\partial x}}\normalsize} \right)}^2} + {{\left( {\large\frac{{\partial z}}{{\partial y}}\normalsize} \right)}^2}} dxdy} \)
    35. Mass of a lamina
      \(m = {\large\iint\limits_R\normalsize} {\rho \left( {x,y} \right)dA},\)
      where the lamina occupies the region \(R\) and its density at a point \({\left( {x,y} \right)}\) is \({\rho \left( {x,y} \right)}.\)
    36. Static moments of a lamina
      The static moment of a lamina about the \(x\)-axis is given by the formula
      \({M_x} = {\large\iint\limits_R\normalsize} {y\rho \left( {x,y} \right)dA} \)
      Similarly, the static moment of a lamina about the \(y\)-axis is expressed in the form
      \({M_y} = {\large\iint\limits_R\normalsize} {x\rho \left( {x,y} \right)dA} \)
    37. Moments of inertia of a lamina
      The moment of inertia about the \(x\)-axis is given by
      \({I_x} = {\large\iint\limits_R\normalsize} {{y^2}\rho \left( {x,y} \right)dA} \)
      The moment of inertia about the \(y\)-axis is determined by the formula
      \({I_y} = {\large\iint\limits_R\normalsize} {{x^2}\rho \left( {x,y} \right)dA} \)
      The polar moment of inertia is equal to
      \({I_0} = {\large\iint\limits_R\normalsize} {\left({x^2} + {y^2}\right) }\) \({\rho \left( {x,y} \right)dA} \)
    38. Center of mass of a lamina
      \(\bar x = {\large\frac{{{M_y}}}{m}\normalsize} =\) \({\large\frac{1}{m}\normalsize} {\large\iint\limits_R\normalsize} {x\rho \left( {x,y} \right)dA} =\) \({\large\frac{{\iint\limits_R {x\rho \left( {x,y} \right)dA} }}{{\iint\limits_R {\rho \left( {x,y} \right)dA} }}\normalsize,}\;\)
      \(\bar y = {\large\frac{{{M_x}}}{m}\normalsize} =\) \({\large\frac{1}{m}\normalsize} {\large\iint\limits_R\normalsize} {y\rho \left( {x,y} \right)dA} =\) \({\large\frac{{\iint\limits_R {y\rho \left( {x,y} \right)dA} }}{{\iint\limits_R {\rho \left( {x,y} \right)dA} }}\normalsize} .\)
    39. Charge of a plate
      \(Q = {\large\iint\limits_R\normalsize} {\sigma \left( {x,y} \right)dA}, \)
      where the electrical charge is distributed over the region \(R\) and its density at a point \({\left( {x,y} \right)}\) is \({\sigma \left( {x,y} \right)}.\)
    40. Average of a function
      \(\mu = {\large\frac{1}{S}\iint\limits_R\normalsize} {f\left( {x,y} \right)dA},\)
      where \(S = {\large\iint\limits_R\normalsize} {dA}.\)