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How can calculus be used to estimate a rocket’s total fuel consumption?

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transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

To estimate a rocket’s total fuel consumption using calculus, one must analyze the rate of mass discharge over time. The total propellant mass consumed, $M_{p}$ , is the integral of the mass flow rate, $\dot{m}$ , over the duration of the burn, $t$ :

$M_{p} = \int_{0}^{t} \dot{m} (t) d t$

In rocket propulsion, the thrust $F$ is defined by the product of the mass discharge rate and the effective exhaust velocity ( $v_{e}$ ), expressed as $F = \dot{m} v_{e}$ . Because the mass flow rate is directly tied to the thrust produced, engineers use this relationship to determine how quickly the propellant supply is depleted.1

For solid-rocket motors, the mass flow rate is determined by the burning surface area ( $S_{c}$ ) of the propellant and the rate at which the surface burns into the solid ( $r$ ), where $\dot{m} = ρ_{p} S_{c} r$ . Since the burning surface area may change over time based on the propellant's geometry, calculus allows designers to integrate these time-varying factors to predict the total fuel consumed throughout a mission. This analysis is essential for optimizing the mass ratios required to achieve specific mission velocities.21