The voltage across an element is 12e^{-2t} V. The current entering the positive terminal of the element is 2e^{-2t} A. Find the energy absorbed by the element in 1.5 s starting from t = 0.

#### Solution:

The energy absorbed can be found by:

\,\displaystyle W=\int^{t_2}_{t_1} (v)(i)\,dt

(Where W is energy absorbed, v is voltage, t is time, and i is current)

Substitute our voltage and current equations:

\,\displaystyle W=\int^{1.5}_{0} (12e^{-2t})(2e^{-2t})\,dt

\,\displaystyle W=\int^{1.5}_{0} (24e^{-4t})\,dt

W=\dfrac{24e^{-4t}}{-4}\Big|^{1.5}_{0}

W=5.985 J

\,\displaystyle W=\int^{1.5}_{0} (24e^{-4t})\,dt

W=\dfrac{24e^{-4t}}{-4}\Big|^{1.5}_{0}

W=5.985 J