Question

how can i find the area of the region enclosed by these curves and line: 4x^2+y=4 and x^4-7=1

Answer 1

actually these are just curves

Answer 2

should i first try and solve for y

Answer 3

r u sure the 2nd equation is right? r u missing a y in there somewhere i mean?

Answer 4

yeah and also draw them to see which id the upper/lower function

Answer 5

*is

Answer 6

the second funtion is x^4-y=1

Answer 7

ok as nadeem said u need to kno the upper and lower function.
also u need to find where they intersect

Answer 8

so i solved for y and got: y=4-4x^2 and y=-1+x^4

Answer 9

huh, the upper curve is y=4-4x^2

Answer 10

good now find where the intersect by setting the equations equal to each other and solve. so solve:
4-4x^2=-1+x^4

Answer 11

If I'm not mistaken 4-4x^2 is you upper function

Answer 12

yeah

Answer 13

they are intesecting parabolas

Answer 14

yeah

Answer 15

what are the intersection points

Answer 16

The intersecting points are -1 and 1

Answer 17

i am trying to solve byy setting them equal to each other but.....

Answer 18

4-4x^2=-1+x^4
x^4+4x^2-5=0
(x^2+5)(x^2-1)
(x^2+5)(x+1)(x-1)
\[x= \pm1\]

Answer 19

you should get :
4(1-x^2)=x^4-1
the LHS will be zero when x=+1,-1 which is the same as the RHS bc of the even powers

Answer 20

so limits of integration are -1 to 1

Answer 21

wait is this symmetry??

Answer 22

you should realize something right of the back, sine both functions are symmetric with the y axis, all you have to do is take the limits on integration fron [0, 1] and multiply the area by 2

Answer 23

yeah

Answer 24

yeah, very nice

Answer 25

makes life much easier for me

Answer 26

lol... sure does

Answer 27

so i got the integration from 0 to 1 of 5-4x^2-x^4

Answer 28

and double that:)

Answer 29

yes

Answer 30

yyup or u can integrate it from -1 to 1 which ever u want

Answer 31

hahaha.... you didn't feel like going through latex

Answer 32

5+5-4/3-4/3-1/5-1/5
=10-8/3-2/5
=104/15

Answer 33

104/15

Answer 34

wow, thanks for that attachtment myiniaya, that was alot of work

Answer 35

and thanks again nadeem

Answer 36

no problem

Answer 37

np

Answer 38

don't worry about what i just typed

Answer 39

lol

Answer 40

lol