Properties of the Definite Integral

Calculus 1 · Axiom Academy

LESSON Properties of the Definite Integral Six rules that let you split, flip, scale, and bound a definite integral — each one just the picture of signed area, rearranged. Multiplying the integrand by a constant c scales every thin strip by c , so the whole accumulated area scales by c . That means the constant can come out in front of the integral sign. the constant scales the area; it never disappears 2. Split a Sum Into Two Integrals Stack the graph of g on top of f and the heights add at every point: the column over each x becomes f(x) + g(x) . So the area under the sum is the area under f plus the area under g — and the same works for a difference. 3. When the Limits Meet, the Integral Is Zero An integral measures area over an interval. Slide the upper limit b back toward the lower limit a and the region narrows to a sliver, then to nothing. With no width there is no area — whatever f is. a single point has zero width, so zero area 4. Swapping the Limits Flips the Sign The integral has a built-in direction. Sweeping from a to b adds strips of positive width dx ; sweeping the other way, from b to a , makes every dx negative. Same region, opposite sign. reversing the direction of travel negates the result 5. Break One Interval Into Two Pick any interior point c and drop a divider. The area from a to b is exactly the area from a to c plus the area from c to b — the pieces simply add back to the whole. adjacent intervals join when the inner limits match

This is the written version of the interactive lesson above. See the full Calculus 1 course.