Multivariate Calculus - Bruce E. Shapiro

LECTURE 17. DOUBLE INTEGRALS OVER GENERAL REGIONS 137
Figure 17.3: The integral over this y-simple region is examined in the example.
Definition 17.2 A region S is called x -simple if there are numbers c, d ∈ R and
functions g(y), h(y) : [c, d] ↦→ R such that
S = {(x, y) : g(y) ≤ y ≤ h(y) and y ∈ [c, d]}
Definition 17.3 A region S is called y-simple if there are numbers a, b ∈ R and
functions f(x), g(x) : [a, b] ↦→ R such that
S = {(x, y) : f(x) ≤ x ≤ g(x) and x ∈ [a, b]}
Figure 17.4: Examples of x-simple and y-simple regions.
Figure 17.5: The quarter of the circle of radius a centered at the origin that is in
the first quadrant.
Example 17.2 Set up the integral ∬ s
f(x, y)dA over the portion of the interior of
a circle of radius a centered at the origin that lies in the first quadrant.
Math 250, Fall 2006 Revised December 6, 2006.