Area under the curve.
The accumulation of quantities; the reverse of differentiation.
Used by Archimedes to estimate the value of Pi by adding up the areas of many tiny polygons.
Add 1 to the exponent, divide by the new exponent. Don't forget + C!
In Plain English: If the Derivative slices a shape into infinite tiny lines to see the slope, the Integral glues them back together to see the whole area. It sums up change over time to give you a total.
In The Real World: Digital Photography. A camera sensor collects (integrates) photons (light particles) over a split second to form a total image brightness.
Forgetting '+ C'. When you reverse a derivative, you don't know if there was a constant number attached, so you must add a generic placeholder 'C'.