Units of an integral: rate times time
About this app
A strip under a rate is height times width, with units on both edges: 5.94 m/s · 1 s = 5.94 m. The strips add up to distance, volume, energy or charge, no longer a rate. Below, the stock with its initial value: the integral is the change. Five contexts from velocity to solar power, plus the quiz “Which unit?”.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The idea
If you integrate a rate over time, the result has the unit of the rate times the unit of time. The integral of a velocity in m/s over time in s is a distance in m, the integral of a power in kW over hours is an energy in kWh, the integral of a current in A over seconds is a charge in C. The integral is the change of the stock: ∫ₐᵇ f(t) dt = B(b) − B(a). You only get the stock itself with the initial value.
What the app shows
At the top you see the rate with labelled axes and n strips below it. One strip is highlighted: on the left its height in the unit of the rate, on top its width Δt in the unit of time, above that the product, for example 5.94 m/s · 1 s = 5.94 m. The header shows the sum of the strips and the exact integral with unit. Below is the stock with its initial value, the staircase of the strip sum and brackets for initial value, change and stock. There are five contexts: velocity, inflow, solar power, current and water level.
Things to try
Answer the quiz “Which unit does the integral have?” and read the reason. Start “Add up”: a time marker runs from a to b, and every strip raises the stock below by its piece. Move the initial value and watch the stock change while the integral stays the same. The water level shows negative strips: when the water falls, the stock decreases, and from 0 to 8 hours the change is zero.
Frequently asked questions
- What unit does an integral have?
- The unit of the function times the unit of the variable of integration. For ∫ v(t) dt with v in m/s and t in s you get m/s · s = m. The reason: every strip of the Riemann sum is f(t) · Δt, and a sum of like quantities has the same unit.
- Why is the integral of velocity a distance?
- In a short piece of time Δt a body moving at velocity v covers roughly the distance v · Δt. Adding up these pieces and making them ever finer gives ∫ v dt, the total change in position in metres. The unit m/s belongs only to v itself.
- Is the integral the stock?
- No, it is the change of the stock between a and b. If a tank held 30 L at the start and 50 L flow in, the integral of the inflow rate is 50 L, but the stock at the end is 80 L. In general B(b) = B(a) + ∫ₐᵇ f(t) dt.
- What can I try in the app?
- You choose one of five contexts, set the interval and the number of strips, tap a strip and read its height, width and product with units. You let the strips add up, move the initial value and answer the quiz about the unit of the integral.
Subject: Analysis (one-dimensional)
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