In Exercises 1-14, evaluate the iterated integral. ∫-1^0 ∫-1^1(x+y+1) d x d y

Name: Problem 3, Chapter 15: Thomas Calculus
Uploaded: 2020-05-06T20:47:40-08:00
Duration: 5 min 12 s
Channel: Emily Frampton
Description: In Exercises 1-14, evaluate the iterated integral. ∫-1^0 ∫-1^1(x+y+1) d x d y

In Exercises 1-14, evaluate the iterated integral. ∫-1^0...

Key Concepts

Linearity of Integration

The linearity of the integral states that the integral of a sum of functions is equal to the sum of the integrals of the functions. This property is useful in simplifying the computation of integrals, as it allows one to break down a complex integrand into simpler parts that can be integrated separately.

Fubini's Theorem

Fubini's Theorem provides the conditions under which a double integral can be computed as an iterated integral. It ensures that, for functions that are either continuous or meet specific integrability conditions on rectangular regions, the order of integration can be interchanged without affecting the result.

Iterated Integral

An iterated integral is the process of evaluating a multiple integral by integrating one variable at a time. This technique simplifies the computation by reducing a multi-dimensional integral into successive one-dimensional integrals, each performed over a specific range corresponding to one variable.

Double Integral

A double integral is used to calculate the accumulation of a quantity over a two-dimensional region. It extends the concept of a single integral to functions of two variables and can represent areas, volumes, and other aggregate measures over a plane.

Transcript

00:01 So today we're going to practice some double integrals, and this is one we're going to start with.

00:06 So this seems a little bit difficult when we first glance at it, but i'm going to try to simplify it, so it's easier to kind of see what we're doing.

00:15 So first thing i'm going to do is this r here.

00:20 Does it have anything to do with this? yes, so i'm going to pull outside of that first integral.

00:27 So i'll have the integral from 0 to 1 of r times.

00:32 So the integral from 0 to 1 of 1 over 1 plus r sdsdr.

00:49 So then, for my previous calculus classes, i know that if i integrate 1 over x, it's the same as the natural log of x.

01:02 So the natural log of whatever is on the bottom times by the derivative of the bottom.

01:07 So here i'll have my integral from 0 to 1 of r times the natural, or not i guess not times natural log of 1 plus r s times that by the derivative, so 1 over r.

01:30 And then i'm still going to have dr.

01:34 And actually before that, i'm going to plug in.

01:36 I'm going to have my bounds.

01:38 So then i'm going to move up to the top right of the page.

01:40 So i have r times 1 over r, which is just going to be 1.

01:46 I'll still have my integral out here.

01:49 And then i'm going to plug in my bounds.

01:51 So i have the natural log of 1 plus r times 1.

01:55 So i'll have the natural log of 1 plus r minus the natural log of 1 plus 0 or the natural log of 1.

02:11 Okay, so the natural log of 1 is 0...