Finding the impulse response from the integral
You are trying to solve for 
$$y(t) = \int_0^1 \! \delta (t-\lambda ) \mathrm{d}\lambda$$

But notice that
$$\int_0^1 \! \delta (t) \mathrm{d}t =1,$$
and also
$$\int_0^1 \! \delta (T-t) \mathrm{d}t = \begin{cases}&amp;1,&amp; 0 \leq T \leq 1\\ &amp; 0,&amp;\text{otherwise.}\end{cases}$$

Since you have that \$T\$ term varying in your equation

$$y(t) = \int_0^1 \! \delta (t-\lambda ) \mathrm{d}\lambda$$
it will depend on \$t\$ the same way as it depended on \$T\$.