1. Home
  2. >Calculus 3
  3. >Multiple Integrals
Help

Double integrals over a rectangular region

Everything You Need in One Place

Homework problems? Exam preparation? Trying to grasp a concept or just brushing up the basics? Our extensive help & practice library have got you covered.

Learn and Practice With Ease

Our proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals.

Instant and Unlimited Help

Our personalized learning platform enables you to instantly find the exact walkthrough to your specific type of question. Activate unlimited help now!

0/2
?
Intros
Lessons
  1. Double Integrals Over a Rectangular Region Overview:
  2. 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
  3. 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
0/0
?
Examples
Lessons
    Topic Notes
    ?
    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=∫ab∫cdf(x,y)dydx \int \int_R f(x,y) dA = \int_a^b \int_c^d f(x,y)dydx
    =∫cd∫abf(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:

    ∫ab∫cdg(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)