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Double integrals over a general region

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Intros
Lessons
  1. Double Integrals Over a General Region Overview:
  2. Double Integrals Over General Regions
    • Not a rectangular region
    • Region is between two curves
    • Case 1: Two curves in terms of xx
    • Case 2: Two curves in terms of yy
    • An Example
  3. Properties of Double Integrals
    • 3 properties
    • Sum of two functions
    • Factoring the scalar
    • Splitting DD into 2 regions: D1D_1 & D2D_2
    • An Example of using the properties
  4. Volume of General Region in 3D
    • Subtract the two functions
    • Find the region DD
    • An Example
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Examples
Lessons
  1. Evaluating the Double Integral Over General Regions
    Evaluate ∫∫Dx−y2dA, \int \int_{D} x - y^{2} dA, \, where DD is the region bounded by  y=12x  \, y = \frac{1}{2}x \, and  y=x \, y = \sqrt{x} .
    1. Evaluate ∫∫D1y dA, \int \int_{D} \frac{1}{y} \, dA, \, where DD is the region bounded by  x=−y  \, x = -y \, and  x=y2 \, x = y^{2} .
      1. Evaluate ∫∫Dxy−x dA, \int \int_{D} xy - x \, dA, \, where DD is the region bounded by  y=x2−1  \, y = x^{2} - 1 \, and  x− \, x-axis.
        1. Evaluate ∫∫De1−x dA, \int \int_{D} e^{1-x} \, dA, \, where DD is the region bounded by the triangle with vertices (0,0), (2, 0), (0, 2).
          1. Find the area of the region bounded by  y=1−x2  \, y = 1 - x^{2} \, and  y=x2−1 \, y = x^{2} - 1.
            Topic Notes
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            Notes:

            Double integrals of a multi-variable function give the volume under the function f(x,y)f(x,y)

            Double Integrals Over General Regions

            Suppose the region is not rectangular, but rather the region is between two curves. For example, we want to integrate within the following region DD, where DD is:

            a≤x≤ba \leq x \leq b
            g1(x)≤y≤g2(x) g_1 (x) \leq y \leq g_2(x)

            Then the iterated integral will be:

            ∫∫Df(x,y)dA=∫ab∫g1(x)g2(x)f(x,y)dydx \int \int_D f(x,y)dA = \int_a^b \int_{g_1(x)}^{g_2(x)}f(x,y)dydx

            Likewise, suppose we have the region DD, where DD is:

            g1(y)≤x≤g2(y) g_1 (y) \leq x \leq g_2(y)
            a≤y≤ba \leq y \leq b

            Then the iterated integral will be:

            ∫∫Df(x,y)dA=∫ab∫g1(y)g2(y)f(x,y)dxdy \int \int_D f(x,y)dA = \int_a^b \int_{g_1(y)}^{g_2(y)} f(x,y)dxdy

            You usually find the region DD yourself.


            Properties of Double Integrals

            The three properties of double integrals are the following:

            ∫∫Df(x,y)+g(x,y)dA=∫∫Df(x,y)dA∫∫Dg(x,y)dA \int \int_D f(x,y) + g(x,y) dA = \int\int_D f(x,y)dA \int \int_D g(x,y) dA
            ∫∫Dcf(x,y)dA=c∫∫Df(x,y)dA,where  c  is  a  constant \int \int_D cf(x,y) dA = c \int \int_D f(x,y) dA, \mathrm{where\;c\; is\;a\;constant}
            ∫∫Df(x,y)=∫∫D1f(x,y)+∫∫D2f(x,y)  where  D  is  split  into  2  regions  D1  &  D2 \int \int_D f(x,y) = \int \int_{D_1} f(x,y) + \int \int_{D_2} f(x,y) \;\mathrm{where\; D\; is\; split \; into\; 2\; regions\; D_1 \; \& \; D_2}


            Volume of General Regions in 3D

            Suppose you want to find the volume of a region that is above g(x,y)g(x,y) and below f(x,y)f(x,y), bounded by a region DD. Then, the volume is:

            ∫∫Df(x,y)−g(x,y)dA\int \int_D f(x,y) - g(x,y)dA