Cambridge O Level Additional Mathematics · Syllabus 4037 · Calculus
Definite Integral
What is Definite Integral?
The number obtained by evaluating an antiderivative at the upper limit and subtracting its value at the lower limit, written as the integral from a to b of f of x with respect to x, and equal to F of b minus F of a. No arbitrary constant appears, because any constant added to the antiderivative cancels in the subtraction. A definite integral measures signed accumulation: where the curve lies above the horizontal axis the contribution is positive and where it lies below the axis the contribution is negative. It therefore equals the geometric area of the region between the curve and the axis only when the curve does not change sign across the interval of integration.
This definition is part of the Calculus chapter in Cambridge O Level Additional Mathematics.
Common mistakes with Definite Integral
- 4. “The integral came out negative, so I made a sign error.” Why it failsA definite integral is a signed accumulation. Where the curve is below the axis it contributes negatively, entirely correctly. A negative value is information, not a mistake. CorrectIf a geometric area is wanted, find every crossing of the axis, integrate between consecutive crossings, and add the magnitudes. Detect itAsk whether the question said “evaluate the integral” or “find the area”. They are different requests.
Questions students ask about Definite Integral
Why is a negative value from a definite integral not automatically a sign error?
Because a definite integral is a signed accumulation: where the curve lies below the horizontal axis, its contribution to \(\displaystyle\int_a^b f(x)\,dx\) is genuinely negative, not a mistake. It only equals the geometric area of a region when the curve does not change sign across the interval. If a geometric area is wanted, find every point where the curve crosses the axis, integrate each piece separately, and add the moduli of the separate values together.