Solve the problem.
-The voltage $\mathrm { v }$ (in volts) induced in a tape head is given by $\mathrm { v } = \mathrm { t } ^ { 2 } \mathrm { e } ^ { 3 \mathrm { t } }$, where $t$ is the time (in seconds). Find the average value of $v$ over the interval from $t = 0$ to $t = 2$. Round to the nearest volt.
A) 6 volts
B) 194 volts
C) 40 volts
D) 1564 volts

Question

Quizplus · Accepted Answer

The average value of a function $f(t)$ over an interval $[a, b]$ is given by $\frac{1}{b-a} \int_{a}^{b} f(t) \, dt$. For $v = t^2 e^{3t}$ from $t = 0$ to $t = 2$, the average value is $\frac{1}{2-0} \int_{0}^{2} t^2 e^{3t} \, dt$. Solving this integral and then dividing by 2 gives an average value of approximately 194 volts, rounding to the nearest volt.

Solve the Problem v\mathrm { v }v (In Volts) Induced in a Tape Head Is Given By Volts)

Solve the Problem $\mathrm { v }$ (In Volts) Induced in a Tape Head Is Given By Volts)