Question

find the area enclosed between the curve y=x^3 and the line y=x

Answer 1

did you sketch the region?

Answer 2

im not sure on how to

Answer 3

you're not sure how to sketch x^3 ?

Answer 4

no i mean i sketched it, i dont know what to do after

Answer 5

say, for example, you're asking about the area enclosed for positive values of x..:

Answer 6

you can see that it's the area under the line x (from 0 to 1) minus the area under x^3 (from 0 to 1)

Answer 7

aka:\[\int\limits_{0}^{1} x-x ^{3} dx\]

Answer 8

can you evaluate that integral?

Answer 9

and then how do we find the anti derivitve of that
hat

Answer 10

the usual way:\[\int\limits_{ }^{ } x^n dx = \frac{ x^{n+1} }{ n+1 }\]

Answer 11

so for \[\int\limits_{ }^{} x^1 dx\]
\[\int\limits_{ }^{} x^1 dx = \frac{ x^{1+1} }{ 1+1}\]

Answer 12

try it for x^3...

Answer 13

x^4/4

Answer 14

yes

Answer 15

so \[\frac{ x^2 }{ 2 } -\frac{ x^4 }{4 } \]

Answer 16

ok?

Answer 17

and whats next?

Answer 18

plug in the limts

Answer 19

for the region on the interval x =0...+inf

( 1/2 -1/4 ) - (0/2 -0/4)

Answer 20

kk