How Much Energy is Dissipated in a Resistor in 0.75 Seconds?

[B3NR4Y]

Homework Statement

The circuit in operates at 60 Hz with E_max = 170 V, and R = 4.5Ω .
How much energy is dissipated in the resistor in 0.75 s?

Homework Equations

P = VI
For a circuit like mine with only a power source and resistor, the current and voltage are in phase, so
V = E _max sin (ωt)
and
$$I = \frac{E_{max} sin(\omega*t)}{R}$$

The Attempt at a Solution

Since I want to know the power dissipated over time, I took an integral $$V_{0} I_{0} \int_{0}^{t} sin(\omega t) dt$$
this should give me the total energy dissipated at time t, it doesn't, and I am not sure why.

 

[rude man]

Homework Statement

The circuit in operates at 60 Hz with E_max = 170 V, and R = 4.5Ω .
How much energy is dissipated in the resistor in 0.75 s?

Homework Equations

P = VI
For a circuit like mine with only a power source and resistor, the current and voltage are in phase, so
V = E _max sin (ωt)
and
$$I = \frac{E_{max} sin(\omega*t)}{R}$$

The Attempt at a Solution

Since I want to know the power dissipated over time, I took an integral $$V_{0} I_{0} \int_{0}^{t} sin(\omega t) dt$$
this should give me the total energy dissipated at time t, it doesn't, and I am not sure why.

You ignored your own expressions for V and I in forming your integral ...

 

[B3NR4Y]

You ignored your own expressions for V and I in forming your integral ...

oh, jeez, it should be
$$V_{0} I_{0} \int_{0}^{t} sin^{2}(\omega t) dt$$ ?

 

[rude man]

oh, jeez, it should be
$$V_{0} I_{0} \int_{0}^{t} sin^{2}(\omega t) dt$$ ?

Mucho better!

BTW the integral is easier if you write it as V_oI_o/ω ∫sin²(ωt)d(ωt) with limits 0 to ωt.
As if I'm not being picky enough, you should also use a dummy variable (like t') in the integral. ∫sin²(ωt')dt' with limits of 0 and t.