anonymous 5 years ago what is the best way to solve a double integral question without drawing the figure we need to integrate over?

1. amistre64

guessing lol

2. amistre64

havent tried doubles yet....

3. anonymous

lol.. but you said you were guessing..

4. amistre64

whats the function or functions to double integrate :)

5. anonymous

yeah giv us the function..

6. amistre64

and make it a hard one lol

7. amistre64

something with an exponent of 3 in it ;)

8. amistre64

and a negative

9. anonymous

let it be a double integral of 4xy-y^3 over the area bounded by the curves y=x^2 n y=x^(1/2)

10. amistre64

if we do the x we get......umm........how do we do doubles anyways?

11. amistre64

is this confined to a single plane right?

12. amistre64

i see the football shape made by the bounds of x^2 and sqrt(x)

13. anonymous

the region bounded by those curves looks like a rugby ball..

14. amistre64

dunno what to do with the 4xy-y^3 function tho..... how does that come into play?

15. amistre64

if we int with respect to y it goes; 2xy^2-y^4/4 right?

16. amistre64

and with respect to x its: 4yx^3/3 +xy^3 ??

17. anonymous

the major problem n which also the 1st step is to find the limits for the double integral for individual x and y.. once u r able to get that.. then its more or less like normal integration..

18. anonymous

how hard is it to find limits solve sqrt(x) = x^2

19. anonymous

x^4 = x x(x^3 -1) =0 x=0, x=1 for real solutions

20. anonymous

when x=0 , y= 0 when x=1, y= 1

21. anonymous

so its just $\int\limits_{0}^{1}\int\limits_{0}^{1} f(x,y) dx dy$

22. anonymous

doesnt matter which order u integrate, in the above I do x first

23. anonymous

so f(x,y) = 4xy-y^3 int f dx = 2x^2 y -xy^3 [ from x=1..x=0] = ( [2y -y^3 ] - [0] ) = 2y -y^3 now we integrate that with respect to y , from y=0 to y=1

24. anonymous

so we get [ y^2 -y^4 / 4 ] from y=1..y=0 = 1- (1/4) = 3/4 //final answer