The voltage in the electrical wiring of a house is given by  `V(t)=170sin(120pit)` Find the average voltage over the half cycle time interval `[0, 1/120]`

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tiburtius | High School Teacher | (Level 2) Educator

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You need to calculate the following integral

`int_0^(1/120)tV(t)`

which comes from formula for expected value.

`170int_0^(1/120)tsin(120pit)dt=|(x=120pit),(dt=dx/(120pi)),(x_0=120picdot0=0),(x_1=120picdot1/120=pi)|=`

`170int_0^pi x/(120pi)sin x dx/(120pi)=170/(120^2pi^2)int_0^pi x sin x dx=`

`|(u=x,dv=sin x dx),(du=dx,v=-cosx)|`

`17/(1440pi^2)(-xcosx|_0^pi+int_0^picosx dx)=17/(1440pi^2)(-xcosx+sinx)|_0^pi=`

`17/(1440pi^2)cdotpi=17/(1440pi)`

` `The average voltage over half cycle time interval `[0,1/120]` is `17/(1440pi)`

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