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Use a CAS double-integral evaluator to find the integrals. Then reverse the order of integration and evaluate, again with a CAS.$$\int_{1}^{2} \int_{0}^{x^{3}} \frac{1}{x+y} d y d x$$

$\approx 0.909543$

Calculus 3

Chapter 15

Multiple Integrals

Section 2

Double Integrals over General Regions

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04:18

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. The area above the x-axis adds to the total.

26:18

In mathematics, a double integral is an integral where the integrand is a function of two variables, and the integral is taken over some region in the Euclidean plane.

03:32

Use a CAS double-integral …

03:17

01:27

02:50

03:50

the given double integral is 1 to 2 zero to x square. One divide by X plus y Do I d. X on? We have to use our graphing calculator to estimate the value of this integral. So we'll use a graphing calculator on. We'll get the estimated value off this interval as 0.9095 Now we have toe right this integral by reversing the order of integration. So for that we have toe sketch of the region of integration. So we know that these other limits for why so why's equals 20 y equals two x square on these other limits for X, that means X is equal to one and X is equal to two. Now will be graphing these equation on a graph people. So let's telegraph ever now. So this is the way exes on this is the X axis on this cope. Or you could say the parable represents Why is it called two X squared on Dhere? X is equal to two on this is X equals to one Onda hair wise equals 20 So the reason off integration will be this region. Now we're rewriting the Dublin trigger by reversing the order of integration. So you get double integral D y dx. Now we're writer limits for that. After take a logical sorry head. It should be one divide way Express Why d x d? Why? We had to reverse the order of integration. So I have to take the strip to find the limit of this integration. I'll be spending this region into two. The region that is reasonable one on region member too now al beating Oriental Strip in both the regions. So here X is we know that gold is why is it called toe exquisite? Begin right Acceptable route. Why so limit would be route? Why? To hair access to so too on de y. The limits for vie varies between these two line this one and this one. So here why equals to one on Dhere? Why is equals to four? No, we'll be writing, though double integral for the second region. So the devil in trickle one divide by X plus one x plus y dx dy way. So the limits for Deeks is that means for X. That means this headaches is one on Dhere. X is equal to do decide. And why varies between the line this one on this hair Bicycles to zero on here. Why close to one? Now again, we'll be using a graphing calculator to find, though estimated value of these two integral on it come out to be as 0.9095 nine judo 95. So this is the final reserved.

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