Double integrals over a rectangular region

Double integrals over a rectangular region

Lessons

Notes:

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

Double Integrals Over a Rectangular Region

If f(x,y)f(x,y) is continuous on a rectangular region R=[a,b]×[c,d]R=[a,b] \times [c,d], then

Rf(x,y)dA \int \int _R f(x,y) dA

gives the volume under the function within that region. The integral is also represented as:

Rf(x,y)dA=abcdf(x,y)dydx \int \int_R f(x,y) dA = \int_a^b \int_c^d f(x,y)dydx
=cdabf(x,y)dxdy = \int _c^d \int_a^b f(x,y)dxdy

These integrals are known as iterated integrals.



Double Integrals of Products

If f(x,y)=g(x)h(y)f(x,y)=g(x)h(y) is continuous on a rectangular region R=[a,b]×[c,d]R=[a,b] \times [c,d], then

Rf(x,y)dA=Rg(x)h(y)dA \int \int_R f(x,y)dA = \int \int_R g(x)h(y)dA

which can be rewritten as:

abcdg(x)h(y)dydx=(abg(x)dx)(cdh(y)dy) \int^b_a \int^d_c g(x)h(y)dydx = (\int^b_a g(x)dx ) ( \int^d_c h(y)dy)

  • Introduction
    Double Integrals Over a Rectangular Region Overview:
    a)
    Double Integrals Over a Rectangular Region
    • Double integral = Volume under f(x,y)f(x,y)
    • Iterated Integral
    • R=[a,b]×[c,d]R=[a,b] \times [c,d] is a rectangle
    • Integrals are interchangeable
    • An Example

    b)
    Double Integrals of Products
    • Product of two functions
    • Move all the x's on one side
    • Move all the y's on one side
    • Integral both and multiply
    • An Example