I just need help with changing the order of integration. I can do the actually integral by myself. Thank you :)
int f(x,y) dydx
where y bounds= 1 to e^x and x bounds= 0 to 4
I thought the new bounds were
int f(x,y) dxdy
x= 0 to log(y) y= 1 to e^4
Please let me know if I'm on the right track thank you:)
If you are finding the area bounded by y=1, y=54.6(approx), x=0, x=4, and f(x,y)=e^x, or any other one-to-one function, and if the limits for
∫e^x dydx
are y=1,f(x,y), x=0,4
then
changing the order could be:
x=e^x to 4, y=1, e^4.
However, it depends on the properties of the function, is it one-to-one, is it invertible, etc.
plot for e^x within the integration limits:
http://img829.imageshack.us/img829/1476/1291263700.png
Oh I see thank you
but the new bounds after you have changed the order shouldn't it be
x= log(y)to 4 --> because I rearranged the bounds from y=e^x to x=log(y)?
Yes, indeed. Don't understand why I didn't see it in the first place.
Awesome thank you :)
Yes, you are on the right track!
When changing the order of integration in a double integral, you need to consider the bounds of integration for both x and y. To do this, you will need to look at the original bounds and determine the range that each variable can take on.
In your case, the original bounds are y from 1 to e^x and x from 0 to 4. To change the order of integration, you will integrate with respect to y first and then with respect to x.
To determine the new bounds for y, you need to look at the original bounds of y. In this case, y goes from 1 to e^x. Since we are integrating with respect to y first, the new bounds for y will correspond to the outer integral.
Since y is going from 1 to e^x, we can rewrite this as y = 1 to y = e^x.
Next, we need to determine the new bounds for x. To do this, we can look at the original bounds of x. In this case, x goes from 0 to 4. Since we are integrating with respect to x second, the new bounds for x will correspond to the inner integral.
Since x is going from 0 to 4, we can rewrite this as x = 0 to x = 4.
Putting it all together, the new order of integration will be:
∫(∫ f(x,y) dy) dx
with the bounds as:
x = 0 to 4
y = 1 to e^x
So your expression:
∫(∫ f(x,y) dy) dx
with the new bounds:
x = 0 to 4
y = 1 to e^x
is correct.