Key takeaway: An integral adds up a continuously changing quantity: the indefinite form gives the family of antiderivatives, and the definite form gives the net accumulated amount, or signed area, over an interval.
Grade 12 – AP Calculus ABAP Calculus AB Unit 6Calculus
Definition of Integral
An integral is the calculus tool for accumulating. If a faucet’s flow rate changes every second, the integral of the rate adds up all the tiny amounts to give the total water. On a graph, that total is the signed area between the curve and the horizontal axis.
There are two forms. One with no limits returns a function plus a constant. One with limits of integration returns a single number. Differentiation and integration are inverse processes.
Formula and notation
Indefinite and definite forms:
\[ \int h(x)\,dx=H(x)+C,\qquad \int_{m}^{n} h(x)\,dx=H(n)-H(m). \]
The symbol \(\int\) is a stretched S for sum, \(h(x)\) is the integrand, and \(dx\) names the variable. Here \(H^{\prime}=h\).
Key facts
- Area above the axis counts as positive and area below it as negative.
- The differential dx is not decoration; it names the variable of integration.
- Sums and constant multiples pass through the integral sign.
- Check an indefinite result by differentiating; you must get the integrand back.
Integral example
Evaluate the indefinite integral \(\int(10x^4-6x+3e^x)dx\).
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Split by sums: integrate each term separately.
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Powers: \(\int 10x^4\,dx=2x^5\) and \(\int 6x\,dx=3x^2\). Exponential: \(\int 3e^x\,dx=3e^x\).
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Combine, keeping the sign on the middle term, and add one constant.
The integral is \(2x^5-3x^2+3e^x+C\).
Common mistake
Watch out
A student computes \(\int x\cdot x\,dx\) as \(\frac{x^2}{2}\cdot\frac{x^2}{2}=\frac{x^4}{4}\), integrating each factor separately.
Why it is wrong
This operation does not split over products. Rewrite \(x\cdot x=x^2\), then \(\int x^2\,dx=\frac{x^3}{3}+C\). Differentiating the student’s \(\frac{x^4}{4}\) gives \(x^3\), not \(x^2\), which exposes the error.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus