Find the area of the region enclosed by the functions
y=x^6
y=x

Question

Find the area of the region enclosed by the functions 
y=x^6
y=x

anonymous · Answer

Easy first you need to find the two areas of inersection

anonymous · Answer

intersection*

anonymous · Answer

Then you integrate by those bounding areas and subtract the two functions

anonymous · Answer

is the area of intersection at 1,1??

lgbasallote · Answer

have you tried graphing this?

anonymous · Answer

yes i just did

lgbasallote · Answer

you can find the intersection by making them equal to one another

$$x = x^6$$
$$0 = x^6 - x$$
$$0 = x(x^5 - 1)$$
the intersection would be x = 0 and x = 1

lgbasallote · Answer

so what is the value of y when x = 0

y = x
y = 0

when x = 1
y = x
y = 1

therefore the intersections are (0,0) and (1,1)

anonymous · Answer

ok so ther is two intersections then

anonymous · Answer

but we use x as the bounds because the y are going to vary because of the functions

lgbasallote · Answer

correct me if im wrong but i think the y = x^6 graph is to the right of y=x right

lgbasallote · Answer

sp it would be $$\huge \int_0^1 (x^6 - x)dx$$

lgbasallote · Answer

so simply integrate/
$$\int_0^1 x^6dx - \int_0^1 xdx$$

anonymous · Answer

yes in the right

anonymous · Answer

No I think it's the other way around 

x-x^6

lgbasallote · Answer

do you get what im doing?

anonymous · Answer

yes i do

lgbasallote · Answer

but x^6 is on the right...right @Romero ?

anonymous · Answer

careful here

lgbasallote · Answer

i dont have a graph so it will help if there is one lol

anonymous · Answer

I see it as x^6 is under x

anonymous · Answer

on the interval $[0,1]$ we have $x^6<x$

anonymous · Answer

http://www.wolframalpha.com/input/?i=y%3Dx%5E6+%2Cy%3Dx

anonymous · Answer

yes on the right

anonymous · Answer

you can imagine with out a graph
what is bigger, $\frac{1}{2}$ or $(\frac{1}{2})^6$?

anonymous · Answer

1/2

lgbasallote · Answer

so i guess it's $$\int_0^1 (x-x^6)dx$$
huh

lgbasallote · Answer

ok i admit i was wrong lol

anonymous · Answer

yeah sure and heck if you are going to ask wolfram just ask it is very accommodating http://www.wolframalpha.com/input/?i=area+between+y%3Dx^6+%2Cy%3Dx wont help on an exam though

anonymous · Answer

so it is not on the right

lgbasallote · Answer

@blopez y=x^6 is to the left

lgbasallote · Answer

i just checked the graph haha

anonymous · Answer

i would use the work "below"

anonymous · Answer

ok

anonymous · Answer

Don't ask wolfram for answers... I really don't trust him -.-

anonymous · Answer

thanks for the help