Question

find the area of the region in the first quadrant that is bounded above by the graph of f(x) = sqrt(x) and below by the x-axis and the line g(x) = x - 2 using integration with respect to y. show steps/explain.

Answer 1

draw it, always
find the points of intersection
once you have all that it is easier to approach the problem

Answer 2

how do i find y upper and lower

Answer 3

ya, i've got a rough sketch like that

Answer 4

looks like 2 and 0

Answer 5

for the y-axis

Answer 6

well... u have an exact graph.. how do i find those values otherwise

Answer 7

find the intercsection

Answer 8

set the two formulas equal

Answer 9

and find both coordinates

Answer 10

\[y^2=y+2\implies y^2-y-2=(y+1)(y-2)=0\]and we don't care about the negative intersect since we're in QI

Answer 11

I solved it for y because it's easier than solving \[\sqrt x=x-2\], and also because we are looking for the y-value anyway

Answer 12

how'd you get y^2 = y + 2

Answer 13

oh... i think i got it

Answer 14

\[\sqrt x=y\implies x=y^2\]\[x-2=y\implies y+2=x\]setting\[x=x\]we get\[y^2=y+2\]

Answer 15

now how do we decide which is our upper and which is our lower
do you know?

Answer 16

0 lower, 2 upper... cause 2 is bigger than 0 and it looks right?

Answer 17

yes, but which function minus which function in the integrand?

Answer 18

right most - left most. so here it's y+2 - y^2

Answer 19

yep

Answer 20

well that's all
happy integrating!

Answer 21

how come i dont have to split it into two separate pieces like in the "with respect to x" version

Answer 22

because I neglected to mention that part :P

Answer 23

lol

Answer 24

oh, no, it;s because there is no "chink" in the "right-left"
like there is in the "top-bottom" way along x

Answer 25

it might be right.. i was just wondering

Answer 26

define chink, lol

Answer 27

sharp corner-type pointy thing
^^^
oxford english standard

Answer 28

haha

Answer 29

meh, well i guess i'll just have to eye-ball it

Answer 30

very rough sketch
as we gio along x we are adding up little rectangles that stretch up and down

Answer 31

... eye-balling it probably not good for bridge building :)

Answer 32

yeah, i gotcha

Answer 33

but when we get to the point we have to use a different bottom....