Find the average current for the time interval. 0 ≤ t ≤ 1/240

Question

The oscillating current in an electrical circuit is as follows, where I is measured in amperes and t is measured in seconds.

I = 4 sin(120𝜋t) + cos(60𝜋t)

Find the average current for the time interval. (Round your answers to three decimal places.)

0 ≤ t ≤ 1/240

Answer 1

Integration provides area under a curve. In this case, the y axis is measured in amperes and the x axis is measured in seconds. Integration units (area under curve) are ampere seconds.

THINK area = width x height, the integral gives you width x height. Divide integral area (ampere seconds) by width (seconds) and you are left with average amperes.. Note the width is 1/240

Average = (1/(1/240) {₀∫^1/240 4 sin (120 π t) dt + ₀∫^1/240cos (60 π t) }dt

= 240{(-1/120π) [4cos (120 π [1/240]) - 4cos(0) + (1/60π)[sin (60 π [1/240] - sin (0)]}

=240{(4)/(120π)[-cos(0.5 π) + cos(0)] + (1/60π)[sin(π/4)-sin(0)]}

=(8/π)[0+1] + (4/π)[(sqrt2)/2 -0]

=8/π + 2sqrt(2)/π

=3.447 amperes

Answer 2

(l(1/240) - l(0) )/ (1/240 - 0) = [4sin(π/2) + cos(π/4) - (4sin0 + cos0)] / (1/240

= 240(4 + √2/2 - 1)