Find the number of bacteria after $12$ hours if the population increases at a rate of $f(t)=e^{0.15t^2}$ million bacteria per hour, using a right-hand sum with $\Delta t=4$.
So I made the following subintervals $[0,4],[4,8],[8,12]$, and since it is mentioned right hand sum, we have to use right end point of interval, so is the answer is value of this expression:
$$4e^{0.15(4)^2}+4e^{0.15(8)^2}+4e^{0.15(12)^2}.$$
Is this the correct way to approach this problem?
Yes, this is correct.
The area can be approximated using three rectangles, and their area is just $\text{width} \times \text{height}$. The width of each subinterval is $4$, and the heights are $f(4)$, $f(8)$, $f(12)$ respectively.
Note that this gives you a terrible approximation for the actual value, which is $\int_0^{12} e^{0.15t^2} \ dt$ by the fundamental theorem of calculus. The right-hand rule gives $\approx 9.61 \cdot 10^9$, but the value of the integral is $\approx 6.84 \cdot 10^8$, which is off by more than a factor of $10$.