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What is an Integral?

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University Course Reader · STEM

The area under the velocity curve is found by evaluating the definite integral between the two time bounds.

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Overview

An integral adds up endless tiny pieces to get a total. Where a derivative measures the rate at each instant, an integral collects those rates back into an amount. On a graph it is the area between the curve and the horizontal axis.
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Overview

An integral is a fancy way of saying add up all the little bits. The derivative asks how fast right now; the integral asks how much in total. On a graph it's just the area under the curve. 😎

A quick take — often all you need.

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Detail

Speed times time gives distance, but only if the speed never changes. When it changes, cut the trip into slices so short the speed barely moves inside each one, then add up the small distance from every slice. As the slices shrink toward zero width, the totals settle on one number: the integral. The same trick measures area under a graph, since each slice is a thin rectangle, and a rectangle's area is width times height. With a start and an end point, the written form ∫f(x)dx becomes one exact number, such as the distance covered between hour one and hour two. Integrating a rate gives back the total and differentiating a total gives back the rate, so the two undo each other, a result known as the fundamental theorem of calculus.
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Detail

Slice the thing thin enough that nothing changes much inside any slice, add the slices, then shrink them toward nothing and watch the total settle. That settled number is the integral. It's the area under a graph because every slice is a skinny rectangle, and rectangles are just width times height. Feed it a rate and it hands back a total: speed becomes distance, water flow becomes litres in the tank, power becomes the electricity bill you've been avoiding eye contact with. Add a start point and an end point and you get one exact number instead of a general formula. Best part: integrating and differentiating undo each other, so if you can do one, you can always check the other. 😎

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Analogy

An electricity meter. Your power draw jumps around all day, yet the meter never bills the highest moment. It multiplies each moment's draw by how long it lasted, adds all the pieces, and that total is your bill.
Deepest level reached.
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Analogy

Your phone's data usage. The speed jumps around all day, but the carrier never bills the fastest moment, it bills the pile that built up by the end of the month. Every second you were online quietly added its own slice to that pile.
Deepest level reached.

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AI explanations may contain errors · Not professional advice

Formal definition — The same term, explained the usual way

The definite integral of a function over an interval is defined as the limit of Riemann sums as the partition norm approaches zero, and represents the signed area bounded by the curve and the axis. The fundamental theorem of calculus establishes integration and differentiation as inverse operations, permitting evaluation via antiderivatives.

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