## IrishBoy123 one year ago order of integration

1. IrishBoy123

show that $$\huge \int_{ 0}^{x} [ \int_{0}^{t} f(p) \ dp] \ dt = \int_{0}^{x} (x − p) \ f(p) \ dp$$ by changing order of integration have a real blind spot on this, simply cannot visualise or draw it.

2. ganeshie8

Hmm so $$x$$ is a constant here.

3. ganeshie8

lets sketch the curves representing the bounds |dw:1438709430741:dw|

4. IrishBoy123

question

5. IrishBoy123

solution

6. ganeshie8

that format files wont open on my system

7. Astrophysics

|dw:1438709761438:dw| |dw:1438709783170:dw|

8. IrishBoy123

sorry i will try and convert it the RHS looks like a convolution from laplace, not that that helps me. the LHS has me seriously befuddled. i was thinking Fubini and double integrals, which i can usually handle, but no idea really

9. IrishBoy123

10. ganeshie8

|dw:1438709868443:dw|

11. IrishBoy123

12. IrishBoy123

thank you @Astrophysics , very kind.

13. Astrophysics

No problem :)

14. ganeshie8

changing order of integration, do we get : $\int\limits_{0}^x\int\limits_0^t f(p) dp\,dt~~=~~\int\limits_{0}^x\int\limits_p^x f(p) dt\,dp$

15. Astrophysics

Yes, that's what the solution shows

16. ganeshie8

you may shoot an arrow and see the starting and leaving curves to setup the bounds : |dw:1438710530659:dw|

17. Astrophysics

|dw:1438710651083:dw| When I used to do double integrals I did the same thing, it's easier to see