Verified answer

40

Is this one question or 5 different ones? Some of them seem to be asking for the integral of |t| with respect to t from -4 to 8, others seem to be asking for the integral of |t| from -2 to 4.

Is either of those the question you are trying to ask?

To integrate an absolute value, split the integral into portions below 0 and portions above 0. Since |t| = t when t >= 0, and |t| = -t when t < 0, then you can say

integral(-4,8) |t| dt = integral(-4,0) |t| dt + integral(0, 8) |t| dt

= integral(-4,0) (-t) dt + integral(0, 8) t dt

And both of those just involve integrating t dt, which you should know how to do.

obviously you have to divide the integral:

∫(-4,0) |t| dt + ∫(0,8) |t| dt

∫(-4,0) |t| dt = ∫(-4,0) (-t) dt = - ∫(-4,0) t dt = - (t^2/2)[-4,0] = - (0-8) = 8

∫(0,8) |t| dt = ∫(0,8) t dt = (t^2/2)[0,8] = (32-0) = 32

answer = 32 + 8 = 40

hope it helps