The integral: accumulation from small changes

Where we are on the map

The derivative broke change into an instantaneous rate; the integral assembles small changes into a total.

Begin with a familiar image

A water meter accumulates flow over time. Even when the rate changes each minute, small volumes add to total consumption.

Precise meaning

The definite integral ∫[a,b]f(x)dx is the limit of sums f(xᵢ)Δx as the partition gets finer. It measures signed accumulation. If F′=f, the fundamental theorem gives ∫[a,b]f(x)dx=F(b)−F(a).

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

interval → partition → height × width → sum → limit → antiderivative → boundary difference

Worked example

For f(x)=2x, an antiderivative is F=x², so ∫[0,3]2x dx=9. Geometry agrees: a triangle of base 3 and height 6 has area 9.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Writing only F(b) and forgetting the lower endpoint is wrong. A definite integral is a change in accumulation, so it requires F(b)−F(a).

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. Evaluate ∫[−1,2](2x+1)dx, show signed parts on a sketch, and separately find the ordinary geometric area between graph and axis.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

When the function is unknown but a law for its change is known, we obtain a differential equation.

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