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Calculus: integrals
Intermediate
Accumulating small contributions, the area under a curve.
x = ∫ v dtAn integral adds up many small pieces: the area under a curve. It undoes differentiation, so position is the integral of velocity and work is the integral of force over distance.
For constant acceleration a, ∫a dt = at + v₀ gives velocity, and integrating again gives x = x₀ + v₀t + ½at².
Definite integrals need limits; indefinite integrals need a constant, which is set by initial conditions.
In more detail
An integral adds up many small contributions, such as the area under a graph of velocity against time, which gives displacement. Integration is the reverse of differentiation.
Real-world uses
- Total work done by a changing force is the integral of force over distance.
- Finding the centre of mass of an irregular object is done by integrating over its mass.
- The distance travelled by a car is the integral of its speed over time, which is the area under a speed-time graph.
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Frequently asked questions
What does the area under a velocity-time graph mean?
It gives the displacement over that time interval.
Why is there a constant in an indefinite integral?
Differentiating a constant gives zero, so the integral is only known up to an added constant until initial conditions fix it.
What is the difference between a definite and an indefinite integral?
A definite integral has limits and gives a number, such as the total distance over an interval. An indefinite integral gives a general function plus a constant.
Related topics in Measurement & Math Tools
Cross product · Triple products · Calculus: derivatives · Partial derivatives · Gradient, divergence, curl · Differential equations in physics