Double Integrals over Bounded Nonrectangular Regions

University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
4
Double Integrals over Bounded Nonrectangular Regions
THEOREM 2 Fubini’s Theorem (Stronger Form) :
University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
5
EXAMPLE 3: Find the volume of the prism whose base is the triangle in the xy-plane bounded by the x-axis and the lines y = x and x = 1 and whose top lies in the plane z = ƒ(x, y) = 3 - x - y.
Solution:
University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
6
Finding Limits of Integration
1. Sketch. Sketch the region of integration and label the bounding curves.
2. Find the y-limits of integration. Imagine a vertical line L cutting through R in the direction of increasing y. Mark the y-values where L enters and leaves. These are the y-limits of integration and are usually functions of x (instead of constants).
3. Find the x-limits of integration. Choose x-limits that include all the vertical lines through R. The integral shown here is
4. To evaluate the same double integral as an iterated integral with the order of integration reversed, use horizontal lines instead of vertical lines in Steps 2 and 3. The integral is
University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
7
EXAMPLE 4: Sketch the region of integration for the integral
and write an equivalent integral with the order of integration reversed.
Solution: The region of integration is given by the inequalities x2 ? y ? 2x and 0 ? x ? 2. It is therefore the region bounded by the curves y = x2 and y = 2x between x = 0 and x = 2
To find limits for integrating in the reverse order, we imagine a horizontal line passing from left to right through the region. It enters at x = y/ 2 and leaves at x = ? . To include all such lines, we let y run from y = 0 to y = 4
The common value of these integrals is 8