The Definite Integral and the Area Under a Curve

For a continuous function, estimate its integral on [a, b] with equal-width rectangles. Width is (b - a)/n. Multiply each sampled function value by that width, then add. Use left endpoints, right endpoints or midpoints. For an increasing nonnegative function, left sums underestimate area and right sums overestimate; decreasing functions reverse this. As the widths tend to zero, the sums approach the exact integral.

An integral counts above-axis regions positively and below-axis regions negatively. For total geometric area, split at roots within the interval, check the sign on each stretch and add the absolute values of the integrals. For y = x - 2 on [0, 3], [x²/2 - 2x] gives -2 on [0, 2] and 1/2 on [2, 3]. The signed total is -3/2; the area is 2 + 1/2 = 5/2.

You can use given integral values without knowing a formula for f. Adjacent intervals add: ∫ from a to c of f = ∫ from a to b of f + ∫ from b to c of f. A constant multiplier stays outside: ∫ of kf = k∫ of f. If the parts are 4 and -1, the whole is 3 and twice the whole is 6. A zero signed total can still contain positive area when opposite contributions cancel.