AboutPaul Klarreich Expertise All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions.
I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.
Experience I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.
Question QUESTION: the double integral of x^3sin(y^3)dydx (the first integral is between 1 and 0 and the second integral is between 1 and x^2)
reverse the order of integration and evaluate the integral.
I dont understand why the order of integration needs to be reversed and im not sure how to approach this problem. any help would be awesome
ANSWER: Questioner: Olivia
Category: Calculus
Private: no
Subject: double integrals
Question: the double integral of x^3sin(y^3)dydx (the first integral is between 1 and 0 and the second integral is between 1 and x^2) reverse the order of integration and evaluate the integral. I dont understand why the order of integration needs to be reversed and im not sure how to approach this problem. any help would be awesome
.................................
Hi, Olivia,
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WARNING: THIS DISCUSSION CONTAINS MATERIAL DIFFICULT TO VIEW ON CERTAIN COMPUTING SYSTEMS. VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.
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Is this what you mean?
{1 {x^2
| | ..... dy dx
}0 }1
If so, then note:
The SECOND integral is from 0 to 1. (Not sure if you mean 0 to 1 or 1 to 0.) It's not the first -- it appears first, but gets done last. It means:
{x=1 {y=x^2
| | ..... dy dx
}x=0 }y=1
And you are supposed to do this:
{x=1 [ {y=x^2 ]
| [ | ..... dy ] dx
}x=0 [ }y=1 ]
The ....., of course, means x^3 sin(y^3), so it is:
So you do the inner integral first, pretending that x is a constant. (That's why we can 'take it out' here.)
When you are done with it, since you substitute for y, all the y's disappear.
Then you do:
{x=1
| x^3(Whatever function of x you got) dx
}x=0
Simple, right? But good luck integrating sin(y^3). I don't know how to do it, either.
So YOU HAVE TO REVERSE THE ORDER OF INTEGRATION to get anywhere.
Now, then, do you want to know how?
Give it a try, and if needed, send me a followup.
---------- FOLLOW-UP ----------
QUESTION: sorry the intergral was:
{1 {1
| | x^3sin(y^3) dy dx
}0 }x^2
i just wanted to see if i am on the right track:
i reversed the order of integration getting:
{1 {sqrt(y)
| | x^3sin(y^3) dx dy
}0 }0
{1
| (sqrt(y)^4sin(y^3))/4 dy
}0
then:
{1
| (y^2sin(y^3))/4 dy
}0
Im not sure how to integrate this function so i dont know what to do next. did i make a mistake somewhere?
Answer QUESTION: sorry the integral was:
{1 {1
| | x^3sin(y^3) dy dx
}0 }x^2
i just wanted to see if i am on the right track:
i reversed the order of integration getting:
{1 {sqrt(y)
| | x^3sin(y^3) dx dy
}0 }0
{1
| (sqrt(y)^4sin(y^3))/4 dy
}0
then:
{1
| (y^2sin(y^3))/4 dy
}0
Im not sure how to integrate this function so i dont know what to do next. did i make a mistake somewhere?
........................................................
Hi, Olivia,
You did fine. That is the same integral I got. But it really shouldn't be difficult.
How about this:
{1
| (y^2sin(y^3))/4 dy << your integral (and mine)
}0
