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Integrate:
\( \left[2x^2 - rac{2}{3}x^3
ight]_0^2 = \left(2(4) - rac{2}{3}(8)
ight) - 0 = 8 - rac{16}{3} = rac{24}{3} - rac{16}{3} = rac{8}{3} \).
So, the enclosed area is \( rac{8}{3} \) square units.
A population of bacteria doubles every 3 hours. If the initial population is 500, how many bacteria are present after 12 hours?
Use exponential growth: \( P(t) = P_0 \cdot 2^{t/T} \), where \( T = 3 \) hours.
\( P(12) = 500 \cdot 2^{12/3} = 500 \cdot 2^4 = 500 \cdot 16 = 8000 \).
Thus, 8000 bacteria are present after 12 hours.
Solve the differential equation \( rac{dy}{dx} = 3y \) with initial condition \( y(0) = 2 \).