I need help; I'm stuck. Integrate from k to -k of abs(2x)dx=18, k>0. What is the value of k?

Question

What is the answer?

Answer 1

Hey Ab.

 

The absolute value is an "even function", which means that it is symmetric about the y-axis. This symmetry gives us a nice little property to work with:

 

∫abs(2x) dx [-k,k]

 

is the same thing as

 

2*∫abs(2x) dx [0,k]

 

Because the graph on the left side of the y-axis is the same as on the right side of the y-axis, you can integrate just one side of the function and multiply it by 2. This works only if your bounds are plus and minus the same number (eg, [-2,2] or [-15,15] or [-20,20] etc). In your case, you're going from -k to +k, so this property works just fine for any value of k.

 

2*∫abs(2x) dx = 2*x²

 

Evaluate this from [0,k]

 

2*(k^2 - 0^2)

 

2*k^2

 

Your question then tells you that this value is equal to 18. So

 

2*k^2 = 18

 

k^2 = 9

 

k = +/- 3

 

Since your question specified that k > 0, that means that it can't be -3

 

k = 3

 

Hope this helps.