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Integral calculus - Double integral

Functions of two variables : f(x,y), f(u,v),....
Double integrals : \(\iint\limits_Rf(x,y)dxdy, \iint\limits_Rg(x,y)dxdy, ...\)

Riemann sum : \(\displaystyle\sum_{i=1}^m \sum_{j=1}^n f \bigg(u_i,v_i \bigg) \bigtriangleup x_i \bigtriangleup y_j\)
Small Changes :\( \bigtriangleup x_i \bigtriangleup y_j\)
Regions of integration : R, S
Polar Coordinates :r, \(\theta\)
Area : A
Surface area : S
Volume of a solid : V
Mass of a lamina : m
Density : p(x,y)
First moments of inertia : \(I_x,I_y,I_0\)
Charge of a plate : Q
Charge density : \(\sigma(x,y)\)
Coordinates of center of mass : \(\overline x, \overline y\)
Average of a function : \(\mu\)

  1. Definition of Double Integral 
    The double integral over a rectangle [a,b] \(\times\)[c,d] is defined to be 
    \(\displaystyle\iint\limits_{[a,b] \times [c,d]} f(x,y)dA=\lim_{\substack{max \bigtriangleup x_i\to0\\max \bigtriangleup x_j\to0}} \sum_{i=1}^m \sum_{j=1}^n f \bigg (u_i, v_j \bigg ) \bigtriangleup x_i \bigtriangleup y_j\)
    Where (\(u_i,v_j\)) is some point in the rectangle 
    \((x_{i-1}, x_i) \times ( y_{j-1}, y_j), and \; \bigtriangleup x_i=x_i-x_{i-1}, \bigtriangleup y_j=y_j-y_{j-1}.\)

      The double integral over a general region R is 
    \(\iint\limits_R f(x,y)dA= \iint \limits_ {[a,b] \times [c,d]} g(x,y)dA,\)
    Where rectangle \([a,b] \times [c,d]\) contains R,
    g(x,y)=f(xx,y) if f(xx,y) is in R and g(x,y)=0 Otherwise.
  2. \(\iint\limits_R[f(x,y)+g(x,y)]dA=\iint\limits_Rf(x,y)dA+\iint\limits_Rg(x,y)dA\)
  3. \(\iint\limits_R[f(x,y)-g(x,y)]dA=\iint\limits_Rf(x,y)dA-\iint\limits_Rg(x,y)dA\)
  4. \(\iint\limits_Rkf(xy)dA= k\iint\limits_Rf(x,y)dA,\)
    Where k i s a constant
  5. if f(x,y) \(\leq\) g(x,y) on R, then \(\iint\limits_Rf(x,y)dA \leq \iint\limits_Rg(x,y)dA.\)
  6. iff f(x,y) \(\geq0\) on R and  \(S \subset R\), then 
    \(\iint\limits_sf(x,y)dA \leq \iint\limits_Rf(x,y)dA.\)
  7. if f(x,y)\(\geq\)0 om R and R and S are non- overlapping regions , then \(\iint \limits_{R\, \cup \,S}f(x,y)dA= \iint \limits_Rf(x,y)dA+ \iint \limits_Sf(x,y)dA .\)
    Here \(R \cup S\) is the union of the regions R and S.
  8. Iterated Integrals and Fubini's Theorem 
    \(\iint \limits_Rf(xx,y)dA= \int\limits_a^b \int\limits_{p(x)}^{q(x)}f(x,y)dy\, dx\)
    for a region of type I,
    R={(x,y)|a\(\leq\)x\(\leq\)b, p(x)\(\leq\)y\(\leq\)q(x) }.

    \(\iint \limits_Rf(x,y)dA= \int\limits_c^d \int\limits_{u(y)}^{v(y)} f(x,y)dx\,dy\)
    for a region of type II,
    R={(x,y)|u(y)\(\leq\)\(\leq\)v(Y) , c \(\leq\) y \(\leq\)d }.
  9. Double Integrals over Rectangular Regions 
    if R is the rectangular region [a,b] \(\times\)[c,d], then
    \(\iint\limits_Rf(x,y)dx\,dy= \int\limits_a^b\bigg ( {\int\limits_c^df(x,y)dy} \bigg)dx=\int\limits_c^d \bigg ( {\int\limits_a^b f(x,y)dx} \bigg)dy.\)
    In the special case where the integrand f (x,y) can be written as g(x)h(y) we have 
    \(\iint\limits_Rf(x,y)dx \, dy= \iint\limits_R g(x)h(y)dx\,dy= \bigg ( { \int\limits_a^b g(x)dx} \bigg ) \bigg ( {\int\limits_c^d h(y)dy }\bigg).\)
  10. Change of Variables 
    \(\iint\limits_Rf(x,y)dxdy= \iint\limits_Sf[x(u,v),y(u,v)] \bigg|{\partial(x,y) \over \partial(u,v) }\bigg|dudv,\)
    Where \(\bigg|{\partial(x,y) \over \partial(u,v) }\bigg| = \Bigg | \substack{{\partial x \over \partial u } \ \ {\partial x \over \partial v } \\ {\partial x \over \partial u } \ \ {\partial x \over \partial v } } \Bigg | \neq 0\) is the jacobian of the transformations \((x,y) \to (u,v),\)and S is the pullback of R which can be computed by \(x=x(u,v), y=y(u,v) \) into the definition of R.
  11. Polar Coordinates
    \(x=r\,cos \theta, \, y = r\, sin\theta.\)
  12. Double Integrals in polar coordinates
    The Differential dxdy for Polar Coordinates is 
    \(dxdy= \bigg | {\partial(x,y) \over \partial(r,\theta)} \bigg |drd \, \theta =rdrd \, \theta.\)
    Let the region R is determined as follows :
    \(0\neq g(\theta)\leq r \leq h(\theta), \alpha \leq \theta \leq \beta , Where \;\; \beta-\alpha \leq 2 \pi.\)
    \(\iint\limits_R f(x,y)dxddy =\int\limits_\alpha^\beta \int\limits_{g(\theta)}^{h(\theta)} f (r\; cos\theta,r\,sin\theta)rdrd\,\theta \;.\)

    If the region R is the polar rectangle given by 
    \(0 \le a \le r \le b, \alpha \le \theta \le \beta, \) where  \(\beta - \alpha \le 2 \pi ,\)
    \(\iint\limits_R f(x,y) dxdy= \int\limits_R^\beta \int\limits_a^b f(r\; cos \theta,r \; sin \theta ) rdrd \theta .\)

  13. Area of a Region 
    \(A= \int \limits_c^d \int\limits_{p(y)}^{q(y)} dx\,dy\) ( for a type II region). 
  14. Volume of a solid 
    \(V= \iint \limits_R f(x,y) dA.\)

    If R is a type II region bounded  by x = a , x = b , y = h(x) ,y = g(x) , then 
    \(V = \iint\limits_Rf(x,y)dA=\int\limits_a^b \int \limits_{h(x)}^{g(x)} f(x,y)dydx.\)
    If R is a type II region bounded  by y = c, y = d , x = q(y) , x = p(y) ,  then 
    \(V = \iint\limits_Rf(x,y)dA=\int\limits_c^d \int \limits_{p(x)}^{q(x)} f(x,y)dydx.\)
    If \(f(x,y) \ge g(x,y) \) over a region R, then the volume of the solid between \(z_1=f(x,y)\) and \(z_2=g(x,y)\) over R is given by 
    \(V= \iint \limits_R[f(x,y)-g(x,y)]dA\).