Thread: Fuel Consumption Question

1. Fuel Consumption Question

The rate fuel is consumed by an aircraft is given by$\displaystyle f(a,s)=10+a+\frac{s^2}{2}$ where $\displaystyle a$ is its forward acceleration and $\displaystyle s$ is its vertical acceleration. How much fuel did the aircraft consume over time interval $\displaystyle [0,2]$ if $\displaystyle (a,s)$ are given by the following parameterized curve:

$\displaystyle a(t)=\{{t }$, $\displaystyle t\in [0,1]$ and $\displaystyle 2-t$,$\displaystyle t\in (1,2]$

AND

$\displaystyle s(t)=\{2$, $\displaystyle t\in[0,1]$ and $\displaystyle 1$,$\displaystyle t\in(1,2]$

2. bump

3. bump

4. let $\displaystyle \psi(t) = (a(t),s(t)) \text{ be a parameterization over } t \in S = [0,1]\cup(1,2]$

$\displaystyle \psi(t)$ is a piecewise function:

$\displaystyle \psi(t) = \left\{\begin{array}{ll} (t,2) & \text{for } t\in [0,1] \\ (2-t,1) & \text{for } t\in (1,2] \end{array} \right.$

$\displaystyle \therefore f\circ \psi(t) = \left\{\begin{array}{ll} 12+t & \text{for } t\in [0,1] \\ \frac{25}{2}-t & \text{for } t\in (1,2] \end{array} \right.$

Then calculate $\displaystyle \int_S f\circ \psi(t) dt$

Edit: I'm not sure if this is right btw. You may have to take the sum of two double integrals over the two different domains.