Graph the integral from negative infinity to time t of the functions in Figure E.18 which are zero for all time t < 0.

Question

Answer 1

Graph the integral from negative infinity to time t of the functions in Figure E-18 which are zero for all time .



This is the integral which, in geometrical terms, is the accumulated area under the function from time to time t. For the case of the two back-to-back rectangular pulses, there is no accumulated area until after time and then in the time interval the area accumulates linearly with time up to a maximum area of one at time . In the second time interval the area is linearly declining at half the rate at which it increased in the first time interval down to a value of 1/2 where it stays because there is no accumulation of area for



In the second case of the triangular-shaped function, the area does not accumulate linearly, but rather non-linearly because the integral of a linear function is a second-degree polynomial. The rate of accumulation of area is increasing up to time and then decreasing (but still positive) until time at which time it stops completely. The final value of the accumulated area must be the total area of the triangle, which, in this case, is one.