Equations · Calculus and analysis
Fundamental theorem of calculus
Integrating a rate of change over an interval gives the total change: integration undoes differentiation.
∫ₐᵇ f′(x) dx = f(b) − f(a)
Aethel says
the integral from a to b of f prime of x, d x equals f of b, minus f of a
What it says
If f′ is the rate at which f changes, adding up that rate over every small step from a to b gives the net change in f between those points.
It is why areas under curves can be found from antiderivatives instead of by summing infinitely many thin strips.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| ∫ₐᵇ | the integral from a to b of | the definite integral over the interval from a to b |
| f′(x) | f prime of x | the derivative of f |
| dx | d x | marks x as the variable of integration |
More on∫Integral
Where you meet it
Calculus courses, and every physics problem that turns a rate into a total.
Questions
How do you say Fundamental theorem of calculus out loud?
∫ₐᵇ f′(x) dx = f(b) − f(a) is read: “the integral from a to b of f prime of x, d x equals f of b, minus f of a.” That is exactly what Aethel says when it meets the formula in a book.
What does ∫ₐᵇ mean in Fundamental theorem of calculus?
∫ₐᵇ (said “the integral from a to b of”) is the definite integral over the interval from a to b.
What does f′(x) mean in Fundamental theorem of calculus?
f′(x) (said “f prime of x”) is the derivative of f.
Can I listen to the equations in my own textbook like this?
Yes. Upload a PDF, EPUB or text file and Aethel reads it aloud with every equation spoken as maths, the way it reads this one, and highlights each sentence as it goes. You can start free.