TheIBAxolotl
Evaluating definite integrals
The fundamental idea: if \(F\) is any antiderivative of \(f\), then
\[\int_a^b f(x)\,dx = \big[F(x)\big]_a^b = F(b) - F(a).\]
The constant \(C\) cancels, so it is left out.
Example. \(\displaystyle\int_1^3 (3x^2 - 2x)\,dx = \big[x^3 - x^2\big]_1^3 = (27 - 9) - (1 - 1) = 18.\)
Signed area
A definite integral counts area above the \(x\)-axis as positive and area below as negative. So the integral is a net (signed) area, not always the geometric area. To find the total area:
- Find where the curve crosses the axis.
- Integrate each piece separately.
- Add the absolute values.
Worked example
Find the area enclosed by \(y = x^2 - 4\), the \(x\)-axis and the lines \(x = 0\) and \(x = 3\).
The curve crosses the axis at \(x = 2\): below on \([0, 2]\), above on \([2, 3]\). Using \(F(x) = \frac{x^3}{3} - 4x\):
- \(\int_0^2 (x^2 - 4)\,dx = \frac{8}{3} - 8 = -\frac{16}{3}\) (below the axis),
- \(\int_2^3 (x^2 - 4)\,dx = (9 - 12) - \left(\frac{8}{3} - 8\right) = -3 + \frac{16}{3} = \frac{7}{3}\).
Total area \(= \frac{16}{3} + \frac{7}{3} = \frac{23}{3}\). Note that \(\int_0^3 (x^2 - 4)\,dx = -3\), which is the net value and not the area.
On a GDC you can get the area in one step by integrating the absolute value of the function.
Area between two curves
If \(f(x) \ge g(x)\) on \([a, b]\), the area between them is
\[A = \int_a^b \big(f(x) - g(x)\big)\,dx,\]
"top minus bottom". It does not matter whether the curves are above or below the axis. The limits are usually the \(x\)-coordinates of the intersection points.
Example. \(y = x\) and \(y = x^2\) meet at \(x = 0\) and \(x = 1\), and \(x \ge x^2\) in between. Area \(= \int_0^1 (x - x^2)\,dx = \frac12 - \frac13 = \frac16\). If the curves cross inside the interval, split at the crossing and swap top and bottom.
Properties worth knowing
- \(\int_a^a f(x)\,dx = 0\).
- \(\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx\).
- \(\int_a^b f\,dx + \int_b^c f\,dx = \int_a^c f\,dx\).
- \(\int_a^b k f(x)\,dx = k\int_a^b f(x)\,dx\) and integrals of sums split.
- \(\int_a^b \big(f(x) + k\big)\,dx = \int_a^b f(x)\,dx + k(b - a)\).
These let you answer "given \(\int_1^5 f = 8\), find \(\int_1^5 (2f(x) + 3)\,dx\)": the answer is \(16 + 12 = 28\).
Common mistakes
- Reporting a negative "area", or letting positive and negative parts cancel.
- Getting the subtraction order wrong in \(F(b) - F(a)\), especially with negatives.
- Using bottom minus top for area between curves.
- Not checking whether curves cross between the limits.
Exam tips
- Sketch the region first, even roughly; it shows where to split.
- Bracket each evaluation: \(\big(F(b)\big) - \big(F(a)\big)\).
- On Paper 2, use the GDC for awkward integrals and state the expression and limits used.