Question

Can someone help me proof that (integral of sen^3 x dx from 0 to pi/4) is minus or equal to (integral of sen^2 x dx from 0 to pi/4) without doing the integrals?

Answer 1

sen is a new function?

Answer 2

what is its meaning?

Answer 3

maybe you mean sine?

Answer 4

sorry, sen is sine in portuguese

Answer 5

lol

Answer 6

sure. because
\[\sin(x)\leq 1\] so
\[\sin^3(x)\leq \sin^2(x)\]

Answer 7

\[\int\limits_{0}^{\frac{\pi}{4}}\sin(x)(1-\cos^2(x)) dx\]

Answer 8

oh without doing integrals

Answer 9

inequality follows right?

Answer 10

no one can do those integrals. they are too hard

Answer 11

satellite73: dont understand "inequality follows right?"

Answer 12

this is easy i think

Answer 13

sine from 0 to pi/4 will always be between 0 and 1

Answer 14

It isn't necessary to do integrals. You just need to show that the cube of any number between 0 and 1 is less than or equal to the square of that same number.

Answer 15

so as you multiply it by itself it gets smaller and smaller

Answer 16

It is a consecuence of sin²(x)= 0.5(1-cos(2x))

Answer 17

am i rite?

Answer 18

It follows from if f(x) <= g(x) for all x between a and b then the integral from a to b of f(x) is less than g(x). In this case f(x) is sin^3(x) and g(x) is sin^2(x).

Answer 19

i mean inequality follows because if you have a positive number less that one, say x. then
\[x^2<x\] for sure

Answer 20

yea squeeze theorem ftw?

Answer 21

Thanks so much for all of you that helped me with that one!

Answer 22

point is that on the interval
\[[0,\frac{\pi}{4}]\]
\[\sin^3(x)\leq \sin^2(x)\] so
\[\int_0^{\frac{\pi}{4}} \sin^3(x)dx \leq \int_0^{\frac{\pi}{4}} \sin^2(x)dx\]