Definite Integral

LHC

The question in my textbook was:

[tex]\int_{0}^{2} x^2 \sqrt{4-x^2} dx[/tex]

I decided to just leave out the lower and upper limits for now, and just solve [tex]\int x^2 \sqrt{4-x^2} dx[/tex].

(It's a bit long, but I assure you I did the work.) Upon making the substitution of [tex]x = 2 \sin \theta[/tex], I got it down to:

[tex]\int x^2 \sqrt{4-x^2} dx = \frac{8}{3} \sin^3 \theta + C[/tex]

Now, I'm transforming it back in terms of x, so [tex]\sin \theta = \frac{x}{2}[/tex]

So, I thought it would make this whole thing:

[tex]\int x^2 \sqrt{4-x^2} dx = \frac{8}{3} \times \frac{x^3}{8}[/tex], so

[tex]\int x^2 \sqrt{4-x^2} dx = \frac{x^3}{3}[/tex]

If I finish up the problem by using the limits of 0 and 2, I get:

[tex]\int_{0}^{2} x^2 \sqrt{4-x^2} dx = \frac{8}{3}[/tex]


But the answer I got from the calculator was around 3.14 (not pi, though). Could someone please tell me where I went wrong? Thanks

Dick

Well, something went wrong in the long part you aren't telling us about. Because the derivative of x^3/3 is DEFINITELY not x^2*sqrt(4-x^2).

nicksauce

Also, FYI the integral should be exactly pi (so says maple).

LHC

Well, I think it's some misunderstanding on my part from the conversion between x and theta.

I used the Mathematica online integrator to verify and its answer was:

[tex]\frac{8}{3} \sqrt{\cos^2 x} \sin^2 x \tan x[/tex]...which, when I simplify (and perhaps this is the part where I'm wrong), I just end up with [tex]\frac{8}{3} \sin^3 \theta [/tex]... which is where I started off =P

Dick

What did you put into the integrator??

LHC

(2sinx)^2*sqrt[4-(2sinx)^2]

Dick

You forgot the part of the integrand that's coming from the dx in your substitution.

LHC

Alright, I shall try to fix it. (Gotta go, it's time for dinner.) Thanks for your help!