Question

how to solve? Help please!(:
Integral problem below.

Answer 1

If f is continious on
\[\int\limits_{0}^{6} f(x) dx= 6\]
evaluate
\[\int\limits_{0}^{2} f(3x)dx\]

Answer 2

Have you ever done a u-substitution?

Answer 3

yeah but I dont really understand how to do it. @tkhunny

Answer 4

Pick u = 3x or x = u/3
This gives du = 3dx or dx = du/3

If we ignore the limits, this easily starts to look like this:

\(\int f(3x)\;dx\;=\;\int [f(3x)/3]\;3\;dx\;=\;\int \dfrac{1}{3}f(u)\;du\)

Okay, now we need to consider the limits. We HAD \(x\in [0,6]\). What are the limits for u?

Answer 5

I have no idea? ;/ @tkhunny

Answer 6

Let's try this...

Rather than drag you through the substitution, how about a little exploration.

Please solve this integral:

\(\int\limits_{0}^{4} x^{2}\;dx\)

Answer 7

@Loser66

Answer 8

@agent0smith

Answer 9

So, you're not going to solve that simple integral? It will lead you to a solution.

Answer 10

I can't answer it much differently...

Make a u-sub for this integral: \[\Large \int\limits\limits_{0}^{2} f(3x)dx\] let u = 3x, so \[\Large \frac{ du }{dx } = 3\]or \[\Large \frac{ du }{ 3 } = dx\] so our integral above becomes \[\Large \int\limits\limits\limits_{?}^{?} f(u)\frac{ du }{3 }\]Now find the limits, using u=3x. When x=0, u=0. When x=2, u=6. Put in the limits, bring the fraction 1/3 out front...\[\Large \frac{ 1 }{ 3 } \int\limits\limits\limits\limits_{0}^{6} f(u)du\]

Answer 11

ok, got you, thank you

Answer 12

Why'd you delete your message? Lol i was halfway through reading it :P

Answer 13

hihihi... shame on me.

Answer 14

you leave your explanation there, ok?

Answer 15

Well the last step is just to use this: \[\Large \int\limits\limits_{0}^{6} f(x) dx= 6\]
and this:
\[\Large \frac{ 1 }{ 3 } \int\limits\limits\limits\limits\limits_{0}^{6} f(u)du\] to find the result.

Answer 16

Yeah, that's pretty much what it is. You can ~verify it using a diagram, using x and 3x as your functions, and finding the areas. The area from 0 to 2 will have 1/3rd the area from 0 to 6.

Answer 17

but the variation x is triple , it should be the same. =6

Answer 18

No, see here for an example:


Find the area in those two triangles.

Answer 19

the y=x gives an area of 18 (half base * height)

y=3x gives an area of 6.