Question

The definite integral of x*(sqrt x^2-400) with an upper limit of 29 and lower limit of 20.
After using U-substitution and plugging in the limits I keep getting 32.078 which is wrong, and apparently the right answer is 3087. What am I doing wrong?!?! Please help!!

Answer 1

∫[20,29] x√(x^2-400) dx
If u = x^2-400 then du = 2x dx and you now have
1/2 ∫[0,441] √u du = 1/2 * 2/3 u^(3/2) = 1/3 u^(3/2) [0,441]
= 1/3 441^(3/2) = 3087

You probably used 21 instead of 21^2 for u

Answer 2

You didn't show your work, so I can't tell where your mistake is,
but your integral should have been
(1/3)(x^2 - 400)^(3/2) , make sure you get that

so form 20 to 29 would give you
(1/3)( 441^(3/2) - 0 )
= (1/3)(9261) = 3087