Definite integral
Compute a definite integral as the area under a curve between limits, with the area shaded on the graph.
What a definite integral is
The definite integral of a function from a to b is the area between the graph of the function and the x-axis over that interval. Area above the axis counts as positive, area below the axis as negative. The integral is thus a kind of "infinite sum" of infinitely thin vertical strips under the curve.
∫ab f(x) dx = area under the curve from a to b
Practical uses of the integral
- Physics: distance is the integral of velocity with respect to time, work is the integral of force with respect to distance.
- Computing areas and volumes of irregular shapes that can't be found with basic formulas.
- Probability: the area under a density curve gives a probability (e.g. the normal distribution).
- Economics: total revenue as the integral of marginal revenue, consumer surplus.
The integral and the derivative are inverse operations
The fundamental theorem of calculus states that integration and differentiation are inverse operations of each other. To see how the area is approximated, take a look at Riemann sums – they show the integral as a sum of rectangle areas.
The calculation uses the numerical Simpson's rule, which is very accurate for smooth functions.