confused (integred problems)

[tinalism]

in question i try to find the area between a line and a curve

first i will find the intersect pts and see where it's bound by

the write a(number on top) S b(number on button)

then write the equation wether is line - curve or curve line

i am confused abt the part

when to take antiderviate? is it applied to all quesions? or not?
i also wnat to know when to use dx and when to use dy

the method will be different but how did u know which one to use

p.s i am not familiar to use dy method

help please~

for example

1.find the area bounded bu the curves y=x^3 -x^2 and y=x^2


i find the intersect (0,0)(2,4)
sorry i dunno how to do the math symbal

2 S 0 x^2-(x^3-x^2) = 2x^2-x^3 take anti


-1/4 x^4 + 2/3 x^3] 2(top) 0(button) =
plug in calculator

fnInt(y1,x,0,2) =1 and 1/3

is the answer correct?


other problems
2.find the area enclosed by the curves g(x)=(x-1)^2 and f(x)=x+1


the answer i have is 3 and 1/3


3.find the area of the region to the right of the curve x=y^2 and to the left of the line y=x-2



4. find the area enclosed by the curves x=f(y)=(y-1)^2 and x= g(y)=y+1




5/find the area bounded byt the curves 2(y-1)^2 =x and (y-1)^2 =x-1



6.find the area bounded by the sine curve from 0 to 4 rad



7. find the area of the region in the first quadrant bounded by the curves y=sinx y=cos x and the y ax-s

 

[Gene]

1. Good job all the way thru.
2. no work shown, but the answer look wrong.
3. If you graph the curves you will see that if you use dx (vertical boxes) you will have to do two integrals. One for x = 0 to 1 where the area is between y = -sqrt(x) and y = +sqrt(x) then another from x = 1 to 4 where the area is between y = x-2 and y = sqrt(x).
If you use dy (horizontal boxes) you can get by with one. The area is between x = y^2 and x = y+2 as y goes from -1 to 2.
The integral is
-1S2 ((y+2)-y^2)*dy