Force-time graph

Check the horizontal axis before you read anything off a graph with force up the side. Time and position give different areas with different units, and the two graphs look identical.

Time on the bottom: the area is impulse. EK 4.2.A.4 says the impulse delivered to a system by a net external force is equal to the area under the curve of a graph of the net external force exerted on the system as a function of time. Units of $\text{N}\cdot\text{s}$.

Position on the bottom: the area is work. EK 3.2.A.5 says work is equal to the area under the curve of a graph of $F$ as a function of displacement. Units of joules. Same shape on the page, different quantity.

Signed, both ways. Area below the time axis counts negative, so a force that reverses partway through delivers two contributions that partly cancel. Adding the areas as magnitudes is the standard misread.

Where the average force comes from. Flatten the curve into a rectangle of the same area over the same interval and its height is $F_{\text{avg}}$. That is what makes $\vec{J} = \vec{F}_{\text{avg}}\Delta t$ exact rather than approximate, and it is why a longer collision means a smaller peak force for the same area.

The graph read backwards. EK 4.2.A.5 gives the reciprocal statement: the net external force exerted on a system is equal to the slope of a graph of the momentum of the system as a function of time.

The rotational twin. EK 6.3.C.4 says the angular impulse delivered to an object equals the area under a graph of net external torque against time.

What that area then does to the system belongs to impulse and to the impulse-momentum theorem.