The integral adds up thin slices. Cut the area under a curve into bars, make the bars thinner, and the total settles on the exact area. Drag the number of bars and watch the estimate close in.
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Travel at a steady 3 km per hour for 4 hours and you cover 3 × 4 = 12 km. That is the area of a rectangle under the speed.
When the speed changes, the total is still the area under the speed graph. That is the idea of accumulation: a total built up from a rate.
A total is the area under its rate.
Cut the area under a curve into bars and add them: a Riemann sum. Under y = x² from 0 to 3, 2 bars give about 3.4, 4 bars about 5.9, 16 bars about 8.2.
The exact area is 9. More bars only get closer to it: the true area never changes. The integral is what the sum settles on as the bars become infinitely thin.
More and thinner bars get closer to the true area, which itself never changes.
Area above the ground adds to the total, and area below takes away. This is signed area, and the definite integral counts it.
For y = x − 2 from 0 to 4, the triangle below the axis is −2 and the one above is +2, so the integral is 0 even though area was covered. Adding it all as positive is the usual mistake.
Area above the ground adds to the total, and area below takes away.
Let F(x) be the running total of area up to x. Its slope at x is the height of the curve there, so F is the antiderivative of f.
For the rate y = x, the running total is F(x) = x²/2. The area from 1 to 3 is then F(3) − F(1) = 4.5 − 0.5 = 4: the total at the end minus the total at the start.
The integral undoes the derivative, so the area between two points is the total at the end minus the total at the start.
A way to add up infinitely many thin slices. The definite integral of a rate gives a total, and it equals the signed area under the graph of that rate.
A definite integral has limits and gives a number, such as an area or a total. An indefinite integral gives a whole family of functions, the antiderivative, written with + C.
Constants have a derivative of 0, so many functions share the same slope. x² + 1 and x² + 5 both have the derivative 2x. The + C stands for any constant. Forgetting it is the usual mistake.
They undo each other. The slope of the running area is the height of the curve, and the area between a and b is F(b) − F(a), the total at the end minus the total at the start.
World 11 of Pip: Math Adventure is about integrals. You fill the reservoir, and learn to find the area under a curve and the total of anything that builds up over time.
It is a math game for iPhone, coming soon to the App Store. Join the list and we will tell you the day Pip is out.