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!!
∫[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
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
To solve the definite integral of the function x*sqrt(x^2-400) with the limits from 20 to 29, we will need to use the technique of u-substitution. It seems you might be making a mistake during the substitution or evaluating the limits.
Let's go through the steps to find the correct solution:
Step 1: Let's start by setting u equal to the expression inside the square root:
u = x^2 - 400.
Step 2: Take the derivative of u with respect to x:
du/dx = 2x.
Step 3: Rearrange the equation from step 2 to solve for dx:
dx = du / (2x).
Now, substitute u = x^2 - 400 and dx = du / (2x) into the integral:
Integral of (x*sqrt(x^2-400)) dx = Integral of (x * sqrt(u)) * (du/(2x)).
Step 4: Simplify the expression and cancel out terms:
We can cancel out the x terms:
Integral of sqrt(u) / 2 du.
Step 5: Integrate the simplified expression:
The integral of sqrt(u) is (2/3) * u^(3/2):
(1/2) * (2/3) * u^(3/2) = u^(3/2) / 3.
Step 6: Substitute u back in:
u = x^2 - 400:
(x^2 - 400)^(3/2) / 3.
Step 7: Evaluate the upper and lower limits:
Let's evaluate the integral from the upper limit, 29, to the lower limit, 20:
[(29^2 - 400)^(3/2) / 3] - [(20^2 - 400)^(3/2) / 3].
[(29^2 - 400)^(3/2) / 3] is approximately equal to 3087.
Therefore, the correct answer to the definite integral of x*(sqrt(x^2-400)) from 20 to 29 is approximately 3087.
It's important to double-check for any arithmetic mistakes, especially when evaluating the limits. Make sure to use a calculator for accurate calculations.