SUNY Geneseo Department of Mathematics

Integration over General Regions, Part 2

Friday, November 4

Math 223 01
Fall 2022
Prof. Doug Baldwin

Anything You Want to Talk About?

(No.)

Integration in Mathematica

Use the Integrate function, which takes as arguments an expression to integrate and lists of bounds for definite integrals, or an expression to integrate and variables to integrate with respect to for indefinite ones.

You can download a notebook that demonstrates various forms of Integrate from Canvas.

Integration over General Regions

Continuing the examples and discussion of section 4.2 that we started Wednesday.

A Curved Region

Integrate xy over the region bounded by the y axis, the line y = 1, and the curve y = x².

F of X and Y equals X times Y. Region in X Y plane right of Y axis, below Y equals 1, and above Y equals X squared

The first, and maybe most important, thing to do is decide what the order of the integrals and their bounds will be. One natural choice is to let x range from 0 to 1, and then at each x value, y ranges from x² to 1. This implies that the integral with respect to x is the outer one, and the one with respect to y, which depends on the value of x, is the inner. (As a rule, integrals whose bounds depend on a variable need to be inside the integral that defines that variable):

Parabolic region on positive side of Y axis and above curve Y equals X squared and an integral over it

Evaluating the integral now requires using single-variable integration rules on each integral in turn. You can use expressions as bounds just like you would constants, namely by replacing the variable of integration anywhere it appears with the expression.

Integral from 0 to 1 of integral from X squared to 1 of X times Y is 1 sixth

What do you get if you reverse the order of integration?

Again, the first thing is to work out the bounds on the variables. Since we’re reversing the order, we know the integral with respect to y will be the outer one, and with respect to x the inner, which means we need to work out the x bounds as expressions involving y. The relevant expression is x = √y, from the original equation y = x².

Parabolic region above positive X axis defined by X equals square root of Y, and integral of integral to root Y of X Y

This version of the integral is arguably easier to evaluate than the original, since the exponents involved are lower:

Integral from 0 to 1 of integral from 0 to square root X of X times Y equals 1 sixth

More than Two Variables

Find bounds for the integral of f(x,y,z) = x + y + z over the region between the xy-, xz-, and yz-planes and the plane x + y + z = 1

F of X, Y, Z equals X plus Y plus Z, and 3 dimensional region between origin and plane X plus Y plus Z equals 1

Follow a similar process to the one in two dimensions, namely pick a variable to start with, then figure out how the range of values for the next variable depends on that one, then how the range of values for the third variable depends on the first two, and so on (if there are more than three variables). We decided to start with x, then y, and finally z. Since the plane that defines the top of the region intersects the xy-plane along the line y = 1 - x, the range of y values at each x is 0 to 1 - x; similarly at each (x,y) point in the xy-plane, the range of z values is from 0 to 1 - x - y:

Integral from 0 to 1 of integral from 0 to 1 minus X of integral from 0 to 1 minus X minus Y of X plus Y plus Z

Notice that I haven’t tried any examples of integrals over circular regions? That’s because they’re much easier to do in polar coordinates.

To see how, please read “Polar Rectangular Regions of Integration” and “General Polar Regions of Integration” in section 4.3 of the textbook.

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