simple calc question?

Question

I thought when i do an integral of a polynomial i can do it term by term, so why am i getting a different answer when i write it in two forms:

integral [0, .2] of (1-x)^2

which is (1-x)^3/3

but when i multiply (1-x )^2 out i get and integrate term by term i get.

x -x^2 + x^3/3

am i wrong about doing these term by term?

i just wanted to add that when you then plug in from [0, .2] you get two different answers.

forget it. I think the book makes a mistake. I hate when that happens.

i got it. It was an algebra mistake at the end. Leave it math PhD's to not understand arithmetic .

that was a waste of an hour

Answer 1

First of all your first answer is wrong: Int((1-x)^2) is -(1-x)^3/3
(because of the -x in the expression)
Evaluate from 0 to 2 to get: -(1-2)^3/3+(1-0)^3/3=
1/3+1/3 = 2/3

Evaluate the second expression from 0 to 2 to get:
(2 - 4 +8/3) - (0) = 2/3

-(1-x)^3/3 does not equal x -x^2+x^3/3 but they differ by a constant. Multipliy it out and you'll see.

Answer 2

The first result should be negative, because choosing u = 1 - x means that du = -dx. So you have -[(1 - x)^3] / 3, which I expanded to (1/3)x^3 - x^2 + x - (1/3). The second result is (1/3)x^3 - x^2 + x, which is identical to the first result when you add C, because all constant terms are absorbed by the constant C. You're only getting a different result with your endpoints because of the missing negative sign. It is computationally sound to do any integral term by term.

Answer 3

I think math nerd has what you're after. I haven't checked the DEFINITE integral to verify that each gives 2/3, but (for sure) your first INDEFINITE integral needs a factor of minus one (and the constant of indefinite integration).

But if you ignore the separate constants of integration and subtract your second indefinite integral from your first indefinite (CORRECTED) integral, you'll get minus 1/3 (which is a constant, just like MN indicated it would be].

You're not wrong doing the integration term by term. When you replace a number by its equivalent (in your case you're replacing the integrand of [(1-x)^2] with [1 - 2x + x^2] ), you'll get an equivalent answer.

Answer 4

ok .
how bout using the "anti chain rule"
just say that ur du = -1 dx
so u already have a dx in ur intergal .. therefore u are adding a -1 ..
so add another -1 outside the intergal sign to make the net addition 1.
therefore u have

-1 int (u)^2 du
-1 (u)3/3
u = 1-x
therefore intergal =
-1(1-x)^3/3

hope this helps
let me know if this is confusing.. ill write up somethng and post it online so u can have a small quick review on integrals.

Answer 5

Nothing wrong at all. You need to evaluate this definte integral at 2 and 0. You'll get -2/3 for each approach.

Answer 6

y = A cos x + B sin x y'= -Asinx +Bcosx [1ST DIFFERENTIATION] y" = -Acosx -Bsinx [2d order differentiation] now y"-y= -Acosx-Bsinx -Acosx-Bsinx = -2Acosx-2Bsinx it truly is likewise equivalent to sinx -2Acosx-2Bsinx=sinx consequently equating coefficient of bothsids -2A=0 or A=0 -2B=a million or B=(-a million/2)

Answer 7

Your first answer should be negative.

On the second part I think you messed up your algebra

Answer 8

figure it out yourself.