Area under a curve
The definite integral of a function f between two numbers a and b adds up the values of f across the interval. Geometrically, it is the signed area between the curve y = f(x) and the x-axis, from x = a to x = b: area above the axis counts as positive, area below as negative. Integrals appear wherever quantities accumulate, from distance traveled to work done and total probability.
The fundamental theorem of calculus
To compute a definite integral, find an antiderivative F, a function whose derivative is f, and subtract its values at the bounds:
∫ f(x) dx from a to b = F(b) − F(a)
For polynomials, the antiderivative follows one simple rule: the antiderivative of xⁿ is xⁿ⁺¹ ÷ (n + 1). The calculator handles any polynomial up to degree 4:
F(x) = p₄x⁵/5 + p₃x⁴/4 + p₂x³/3 + p₁x²/2 + p₀x
Worked example
Integrate f(x) = 3x² − 2x + 1 from 0 to 2:
- Antiderivative: F(x) = x³ − x² + x.
- F(2) = 8 − 4 + 2 = 6. F(0) = 0.
- Definite integral: 6 − 0 = 6.
- Average value of f on the interval: 6 ÷ 2 = 3.
- f(0) = 1, f(1) = 2 and f(2) = 9, so Simpson’s rule gives 2 ÷ 6 × (1 + 4 × 2 + 9) = 6, exactly right because the polynomial has degree 2.
Properties worth knowing
- Reversing the bounds changes the sign: ∫ from b to a = −∫ from a to b.
- Splitting: ∫ from a to c = ∫ from a to b + ∫ from b to c.
- Linearity: the integral of a sum is the sum of the integrals, and constants factor out.
- Odd functions integrated over a symmetric interval such as −2 to 2 give 0, because the areas above and below cancel.
Signed area versus total area
If the curve crosses the x-axis, the definite integral subtracts the parts below the axis. For example, x from −1 to 1 gives 0 even though the shaded regions have a total area of 1. To get the total geometric area, find where the function equals zero, integrate each piece separately, and add the absolute values.
Average value
The average value of f on [a, b] is the integral divided by b − a. It is the height of a rectangle with the same base and the same area as the region under the curve. If f is a speed, the average value is the average speed over the time interval.
Numerical methods
For functions without a simple antiderivative, integrals are estimated numerically. Simpson’s rule fits a parabola through the two endpoints and the midpoint: (b − a) ÷ 6 × (f(a) + 4f(m) + f(b)). It is exact for polynomials up to degree 3, which makes it a good check on the exact result for cubic and lower functions. For degree 4 it is close but not exact; splitting the interval into more pieces improves the estimate.