If integrating |2x|dx from -k to k equals 18 and k>0, then what is the value of k?
many thanks!
integral | 2 x | dx =
2 integral | x | dx =
2 ( x ^ 2 ) * sgn ( x ) + C =
integral | 2 x | dx from - k to k =
k ^ 2 - [ ( k ^ 2 ) * ( - 1 ) ] =
k ^ 2 + k ^ 2 =
2 k ^ 2
2 k ^ 2 = 18 Divide both sides by 2
k ^ 2 = 9
k = sqrt ( 9 ) = ± 3
integral | 2 x | dx from - 3 to 3 = 18
k = 3
2 integral | x | dx =
2 ( x ^ 2 ) / 2 * sgn ( x ) + C =
( x ^ 2 ) * sgn ( x ) + C =
To find the value of k, we need to solve the integral ∫|2x|dx from -k to k, which is equal to 18.
First, let's split the integral into two parts since the integrand, |2x|, changes sign at x = 0:
∫|2x|dx = ∫2x dx, for 0 ≤ x ≤ k, and ∫-2x dx, for -k ≤ x < 0.
Integrating, we get:
∫2x dx = x^2, for 0 ≤ x ≤ k,
∫-2x dx = -x^2, for -k ≤ x < 0.
Now, we can calculate the integrals over the given limits:
∫|2x|dx = ∫2x dx + ∫-2x dx = (x^2) from 0 to k + (-x^2) from -k to 0.
For the first part, we substitute the upper and lower limits:
(x^2) from 0 to k = (k^2) - (0^2) = k^2,
For the second part, we substitute the upper and lower limits:
(-x^2) from -k to 0 = -(0^2) - ((-k)^2) = -k^2.
Adding the two parts gives us:
k^2 - k^2 = 0.
Since the integral is equal to 18, we have the equation:
0 = 18.
However, this equation has no solutions, which means our initial assumption is incorrect. Thus, there is no value of k that satisfies the integral ∫|2x|dx from -k to k = 18.