An integral adds up infinitely many tiny pieces to find a total, such as the area under a curve. The definite integral ∫baf(x) dx is the signed area between the graph of f and the x-axis from x = a to x = b. An indefinite integral is an antiderivative: ∫ 2xdx = x2 + C.
A definite integral is the area above the axis minus the area below it, which is why it is called the net signed area.
Formal definition and notation
For a function f on [a, b], the definite integral ∫baf(x) dx is the limit of Riemann sums Σf(xi) Δ x as the widths Δ x of the subintervals shrink to 0, provided the limit exists (it does for every continuous f). It equals the net signed area: area above the x-axis minus area below. The indefinite integral ∫f(x) dx = F(x) + C is the family of all antiderivatives, the functions whose derivative is f. The Fundamental Theorem of Calculus links the two: ∫baf(x) dx = F(b) − F(a).
Symbol
Read it as
Meaning
∫baf(x) dx
the integral from a to b of f of x, d x
a definite integral: a number, the net signed area from a to b
∫f(x) dx
the integral of f of x, d x
an indefinite integral: a family of functions F(x) + C
dx
d x
shows the variable of integration and stands for the tiny width of each slice
A definite integral gives a number; an indefinite integral gives a function plus C.
Area below the x-axis counts as negative, so an integral can be 0 or negative even when the shaded region is large.
Integrating a rate gives a total change: the integral of velocity is distance traveled (displacement), and the integral of a flow rate is the amount that flowed.
The integral of a constant is a rectangle: ∫51 3 dx = 3 × 4 = 12.
Integration and differentiation undo each other: the derivative of ∫f(x) dx is f(x).
Fundamental Theorem of Calculus
Fundamental Theorem of Calculus
∫abf(x) dx = F(b) − F(a)
F
any antiderivative of f, so F′(x) = f(x)
a, b
the lower and upper limits of integration
power rule
∫xndx = xn + 1n + 1 + C for n ≠ −1
Example: ∫31 2xdx = 32 − 12 = 8, because x2 is an antiderivative of 2x.
A derivative measures a rate of change (slope of a curve); an integral adds up a rate to get a total (area). They are inverse operations: the derivative of x3 is 3x2, and the integral of 3x2 is x3 + C.
An integer is a whole number such as −3 or 7. The words look alike, but integral (calculus) has nothing to do with integers.
Quick check: 3 questions
Question 1 of 3
Which is ∫ 2xdx?
Show answer and explanation
Answer: C. The derivative of x2 is 2x, so the antiderivative is x2 + C. 2x2 forgets to divide by the new exponent, 2 is the derivative of 2x, and x2 alone leaves out the constant.
Question 2 of 3
What is ∫31 4 dx?
Show answer and explanation
Answer: A. The region is a rectangle of height 4 and width 3 − 1 = 2: area 8. 4 is only the height, 12 uses width 3, and 16 uses width 4.
Question 3 of 3
The graph of f lies above the x-axis from x = 0 to 2, enclosing area 5, and below it from x = 2 to 6, enclosing area 2. What is the integral of f from 0 to 6?
Show answer and explanation
Answer: D. Area above counts plus and area below counts minus: 5 − 2 = 3. 7 adds the areas (that is total area), −3 subtracts the wrong way, and 2 is only the lower piece.
Integral: frequently asked questions
What is an integral in simple words?
An integral is a way to add up a quantity that keeps changing. Slice the region under a curve into very thin rectangles, add their areas, and let the slices get infinitely thin: the result is the integral. It turns a rate, such as speed, into a total, such as distance traveled.
What is the difference between a definite and an indefinite integral?
A definite integral has limits, like ∫20 3x2dx, and its answer is a number, here 8. An indefinite integral has no limits, like ∫ 3x2dx, and its answer is a family of functions, x3 + C. The Fundamental Theorem of Calculus uses the second to compute the first.
Why do you add C to an indefinite integral?
Because many functions have the same derivative. x3, x3 + 4 and x3 − 10 all have derivative 3x2: the constant disappears when you differentiate. Writing + C records every possible antiderivative at once. In a definite integral the C cancels, so you can leave it out there.
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