1. Fuel Consumption Question

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

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

AND

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

2. bump

3. bump

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

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

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

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

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

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.