Question

Evaluate the integral by reversing the order of integration

Answer 1

\[\int\limits_{0}^{\sqrt{\pi}}\int\limits_{y}^{\sqrt{\pi}}Cos(x^2)dxdy\]

Answer 2

Any bright ideas anoyone?
Don't worry about the integration part, I'll handle that part myself, just need to understand how to rearrange the integrand and change it's limits / boundaries

Answer 3

can you draw the region of integration?

Answer 4

Not sure if this is the way to approach it

Answer 5

\[\int\limits\limits_{0}^{\sqrt{\pi}}\int\limits\limits_{\color{red}{y}}^{\sqrt{\pi}}Cos(x^2)dxdy\]

you have missed out the y i've marked in red, can you include that?

Answer 6

I don't know honestly haha

Answer 7

hint perhaps?

Answer 8

Isn't y, just on the y-axis, is it just on the x-axis?

Answer 9

ok here is how you should actually read the given integral
\[\int\limits\limits_{y=0}^{y=\sqrt{\pi}}~~\int\limits\limits_{x=y}^{x=\sqrt{\pi}}Cos(x^2)dxdy\]

we usually skip writing 'x=' and 'y=' because it is understood and putting it in is just cluttering the whole thing.

Answer 10

so what you have missed out on your graph is x=y
can you put that ?

Answer 11

Ah Okay, it's a lot better to see it like this

Answer 12

Something like this?

Answer 13

yes, you have graphed all the relevant boundaries.
now you need to identify the 'region of integration'

Answer 14

have you learnt how to do that?
perhaps you would have been told since there is a 'dx' first, you need to imagine a 'horizontal strip'
sound familiar?

Answer 15

No I don't think I'm familiar with this

Answer 16

We mainly deal with the limits associated with dx, am i correct? Cause I only cover one question like this in class

Answer 17

I think it depends doesn't it, on which to integrate first whether it is dxdy or dydx based on Fubini thm?

Answer 18

if you are going to change dxdy to dydx, you need to change the limits.

Answer 19

Ah okay, so for this case, we'' just stick to how it is current presented as dxdy then

Answer 20

the question says reverse the order...

Answer 21

in any case, you cant integrate in the given order even if you wanted to
because we dont know integral of cos(x^2)

Answer 22

In regards to determining the region for dx, I know that we have to change the boundaries based on interpreting the graph:

So I would it be this:
x^2 = (pi)
And since x = y
It would be (y^2)

So the Integral would look something like this I suppose:
\[\int\limits_{0}^{\sqrt{\pi}}(\int\limits_{y^2}^{\pi}Cos(x^2)dx)dy\]

Answer 23

Seems a bit weird ^

Answer 24

its incorrect,
first you need to identify the 'region' of integration
and then you need to reverse the order to dydx
when you reverse the order, it needs to be accompanied by appropriate change in the limits

Answer 25

here's how you identify the region of integration

since in the question you have 'dx' first,
imagine a horizontal strip anywhere on the graph, like so