SUNY Geneseo Department of Mathematics

Introduction to Multivariable Integrals

Friday, April 7

Math 223
Spring 2023
Prof. Doug Baldwin

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Lagrange Multipliers

Finish the Cobb-Douglas function example from Wednesday.

p(x, y) = 2 x0.4 y0.6 with the constraint that x + y = 10

Wednesday we calculated the gradients we’d need, but didn’t get further:

Gradients of 2 times X to the 0.4 times Y to the 0.6 and X plus Y

The next thing to do is turn those gradients (and the constraint) into a system of equations to solve:

0.8 X to the negative 0.6 times Y to the 0.6 and 1.2 X to the 0.4 times Y to the negative 0.4 both equal lambda

To solve this, notice that the first two equations imply that 2 different expressions involving x and y are equal to each other. Use this to express y as a function of x, then plug that result into the constraint to solve for x and then y:

Gradient equations imply Y equals 3 halves of X from which constraint gives X equals 4 and Y equals 6

Plugging these values for x and y back into the p function with the help of a calculator, we get that p(4, 6) ≈ 10.2. Assuming that x and y can’t be negative (since they measure inputs to some production process), this value must be a maximum because towards the endpoints of the ranges for x and y (namely 0 to 10), p(x, y) goes to 0.

Introduction to Multivariable Integration

Motivational example: what’s the volume of the region below the surface z = 4 - x2 - y2 and between the planes x = 0, x = 1, y = 0, y = 1?

Adapting ideas from areas under a curve in calculus 1, we could maybe divide the volume up into tall thin boxes, whose individual volume we can calculate.

Box of width delta X and depth delta Y extending from X Y plane up to the surface of a function's graph

Then the whole volume is the sum of the volumes of all the tall thin boxes. And in the limit, as the boxes shrink towards 0 width and depth, we get the exact volume:

V is limit as delta X goes to 0 of limit as delta Y goes to 0 of sums of F of X I and Y I times delta X and delta Y

Just like in single-variable calculus, the limit of the sums defines the definite integral for 2 variables. Analogous definitions work for more variables, too. There are 2 ways people write and talk about these integrals: as fairly abstract integrals over multiple dimensions (“multiple integrals”) and in a more concrete form where the multiple limits of multiple sums are recognized as being integrals of integrals (“iterated integrals”):

Limit of sums is integral of integral of F of X and Y or double integral of F over some region with respect to area

Here’s what this looks like in the example we started with. We start by evaluating the innermost integral, treating every variable except the one it integrates with respect to as a constant (just like with partial derivatives):

Integrating 4 minus X squared minus Y squared from 0 to 1 in Y and then 0 to 1 in X gives 10 over 3

Problem Set

Problem set 10, mostly on optimization but also with an introductory integration question, is ready.

Work on it next week, grade it the week after.

Next

More about iterated integrals.

Please read “Properties of Double Integrals,” and “Iterated Integrals” in section 4.1 of the textbook, and “Integrable Functions of Three Variables” in section 4.4.

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