What Is an Integral?
Where a derivative asks how fast something is changing, an integral asks how much has accumulated. Geometrically that is the area under a curve, and physically it is the total that a rate has built up.
Accumulation, not just area
Slice the region under a curve into thin rectangles, add their areas, and let the slices get thinner. The limit of that sum is the integral. Area is the picture, but the idea is accumulation: if the curve is a speed, the area is a distance.
Indefinite versus definite
- An indefinite integral has no bounds and produces a family of functions, which is why it ends in plus C.
- A definite integral has bounds and produces a single number.
- For a definite integral, evaluate the antiderivative at the top bound and subtract its value at the bottom.
Why plus C
Differentiating destroys constants: x^2 + 7 and x^2 - 3 both differentiate to 2x. Reversing the process cannot recover which constant was there, so every indefinite integral carries an unknown constant. In a definite integral the constant cancels in the subtraction, which is why it is never written there.
Worked examples
Example 1
Question
Find the indefinite integral of 2x with respect to x.
Answer
x^2 + C
Steps
- 1Ask which function differentiates to 2x.
- 2The power rule in reverse: raise the exponent by one and divide by the new exponent.
- 32x becomes 2x^2 / 2 = x^2.
- 4Add the constant of integration: x^2 + C.
- 5Check by differentiating: the derivative of x^2 + C is 2x.
Example 2
Question
Evaluate the definite integral of 2x from 0 to 3.
Answer
9
Steps
- 1Find the antiderivative first: x^2.
- 2Evaluate at the upper bound: 3^2 = 9.
- 3Evaluate at the lower bound: 0^2 = 0.
- 4Subtract: 9 - 0 = 9.
- 5No constant appears, because it would cancel in the subtraction.
Geometrically this is the area of a triangle with base 3 and height 6, which is also 9. The two methods agree, as they must.
Example 3
Question
A car's velocity is v(t) = 6t m/s. How far does it travel in the first 4 seconds?
Answer
48 metres
Steps
- 1Distance is the accumulated velocity, so integrate v(t) from 0 to 4.
- 2The antiderivative of 6t is 3t^2.
- 3At the upper bound: 3(4)^2 = 48.
- 4At the lower bound: 3(0)^2 = 0.
- 5Subtract: 48 - 0 = 48 metres.
Common mistakes
Leaving out plus C on an indefinite integral.
Indefinite integrals always carry it. It is a mark in most mark schemes.
Writing plus C on a definite integral.
It cancels in the subtraction, so it never appears in a definite answer.
Subtracting the bounds in the wrong order.
Upper minus lower, always. Reversing them flips the sign of the answer.
Assuming a definite integral is always the physical area.
Below the axis the integral contributes negatively. For total area, split at the crossings and take absolute values.
Practice it
Work these out first, then open the answer. Stuck halfway? Select the problem and ask Solvecoon for the step you are missing.
1. Integrate 3x^2 with respect to x.
Show answer
x^3 + C. Check by differentiating.
2. Evaluate the integral of 4x from 1 to 2.
Show answer
6. The antiderivative is 2x^2, so 8 - 2 = 6.
3. Integrate (x + 1) with respect to x.
Show answer
x^2/2 + x + C.
Questions people ask
- Is an integral just area?
- Area is the picture, accumulation is the meaning. Integrating a speed gives distance, and integrating a rate of flow gives volume.
- How do I check an integral?
- Differentiate your answer. If you do not get back the original function, the integration is wrong, and this check takes seconds.
- Why does the constant not matter for definite integrals?
- Because it appears at both bounds and cancels when you subtract, so it can never affect the number.
- Can an integral be negative?
- Yes, when the curve is below the axis over that interval. That is correct behaviour, not an error.
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