Integral rules: putting areas together
About this app
The integral as an oriented area balance and its four rules: additivity over intervals with a draggable intermediate point c (even outside [a, b]), swapping the limits with a negative strip width dx, the factor rule with a slider r and the sum rule with f, g and f + g. Below, every sub-area is an arrow: areas below the axis count negative.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The idea
The integral ∫ₐᵇ f(x) dx is an oriented area: areas above the x-axis count positive, areas below count negative, and integrating from right to left flips every sign. Four rules follow. Additivity: ∫ₐᶜ f dx + ∫ᶜᵇ f dx = ∫ₐᵇ f dx. Swapping the limits: ∫ₐᵇ f dx = −∫ᵇₐ f dx, and ∫ₐᵃ f dx = 0. Constant factor rule: ∫ₐᵇ r·f(x) dx = r·∫ₐᵇ f(x) dx. Sum rule: ∫ₐᵇ (f + g) dx = ∫ₐᵇ f dx + ∫ₐᵇ g dx.
What the app shows
At the top you see the graph with the sub-areas between the limits. A sub-area with a positive contribution appears in the course colour, one with a negative contribution in the contrast colour. Below, the balance shows every sub-area as a bar on a number line, to the right or to the left, and an arrow up to the value of the integral. Four modes belong to the four rules: combining intervals with an intermediate point c, swapping the limits with numbered strips and strip width dx, a factor with slider r, and a sum with three cards for f, g and f + g. You can choose x² − 2x, sin x, 0.5x³ − 2x and x + 1.
Things to try
Drag the intermediate point c beyond b: the second integral then runs backwards, the overhanging piece is subtracted again and the sum stays the same. Swap a and b with one click: the same area changes colour, dx becomes negative and the integral changes sign. Set sin x from 0 to 2π: the integral is 0, but the area without sign is 4. In factor mode you stretch f by r and watch the balance grow with it; in sum mode the arrows of f and g add up to the arrow of f + g.
Frequently asked questions
- Why does the sign change when you swap the limits?
- Because you then integrate in the opposite direction. If you cut the area into strips of width dx = (b − a)/n, dx is negative for b < a, and every contribution f(x)·dx changes sign. That is why ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx.
- Is an integral the same as an area?
- No. The integral is an area balance in which areas below the x-axis count negative. ∫₀^{2π} sin x dx = 0, although the region between graph and axis has area 4. For the area you integrate from zero to zero and add the absolute values.
- Does c have to lie between a and b in the additivity rule?
- No. With oriented integrals, ∫ₐᶜ f dx + ∫ᶜᵇ f dx = ∫ₐᵇ f dx holds for any position of c, as long as f is integrable on the whole range. If c lies to the right of b, the second integral runs backwards from c to b and subtracts the overhanging piece again.
- What can I adjust in the app?
- You choose one of the four rules and one of four functions, drag the limits a, b and c in the graph, swap a and b with a button, set the factor r or pick a second function g. The “Values” panel shows the rule with numbers, the positive and negative contributions and the area without sign.
Subject: Analysis (one-dimensional)
More from Analysis (one-dimensional)
- Derivative as a sequence of secants
- Bounded growth
- Units of an integral: rate times time
- Exponential growth and decay
- Optimisation: the largest rectangle
- Area between curves
- Area function is an antiderivative
- Sequential continuity
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