Applying Properties of Definite Integrals

A definite integral is built from a Riemann sum: slice the interval into pieces of width delta x, multiply each by a height f(x), add them up. That picture explains every property here. Additivity says integral from a to c of f plus integral from c to b of f equals integral from a to b of f, as long as c sits between a and b — you're just regrouping the same rectangles into two batches instead of one. The constant-multiple property says integral of k times f equals k times the integral of f, since pulling a constant out of a sum of k times f(x) times delta x just factors it out front.

The sign flip is the one that feels strange until you track direction. Integral from a to b of f equals negative integral from b to a of f, because reversing the limits reverses which direction you walk across the interval, and that flips the sign of delta x in every rectangle. Push that idea to its edge: if a equals b, the interval has zero width, every delta x is zero, and the integral is 0 no matter what f is doing.

These properties are the toolkit you lean on once the Fundamental Theorem of Calculus turns integrals into antiderivative subtraction. Breaking an interval apart, reversing limits, or factoring out a constant lets you handle piecewise functions, combine partial information about an integral, or simplify an expression before evaluating. None of this works for products or quotients of functions — integral of f times g is not integral of f times integral of g, because a Riemann sum for f(x)g(x) is built from the product of heights at each point, not from multiplying two separate sums term by term.