20. The Definite Integral and Area Under a Curve

Starting point: Years 9 and 10

Area formulas

$rectangle: A = l w triangle: A = \frac{1}{2} bh trapezium: A = \frac{h}{2} (a + b)$

Area under a velocity graph is distance

If a car travels at 20 m/s for 10 s, the velocity-time graph is a horizontal line, and the area under it is $20 \times 10 = 200$ m, the distance travelled.

Estimating area under a curve

For a curve, split the area into thin strips and add them up. Using rectangles:

Four strips of width 1. Each rectangle uses the height at its left edge.

The rectangles here use the height at the left of each strip: $1 + 1.5 + 3 + 5.5 = 11$ . The true area is a little more, because the curve rises above each rectangle. Thinner strips give a better estimate.

Area under a curve

Shapes

Rectangle, triangle, trapezium

Estimate

Thin strips
Trapezoidal rule

Exact

Definite integral
F(b) - F(a)

Sign

Above axis: positive
Below axis: negative
Split at roots for area

Meaning

Velocity area: distance
Rate area: total change

Year 12The definite integral and the fundamental theorem

The exact area under $y = f (x)$ from $x = a$ to $x = b$ (with $f \geq 0$ ) is the definite integral:

$\int_{a}^{b} f (x) d x$

The fundamental theorem of calculus links this area to anti-derivatives. If $F$ is any primitive of $f$ :

$\int_{a}^{b} f (x) d x = [F (x)]_{a}^{b} = F (b) - F (a)$

Example. The curve above is $y = \frac{1}{2} x^{2} + 1$ .

$\int_{0}^{4} (\frac{1}{2} x^{2} + 1) d x = [\frac{x ^{3}}{6} + x]_{0}^{4} = \frac{64}{6} + 4 = 14 \frac{2}{3}$

The trapezoidal estimate of 15 was close.

Signed area. Area below the x-axis counts as negative in a definite integral. $\int_{0}^{2 π} sin x d x = 0$ because the part above cancels the part below. To find the actual area, split at the x-intercepts and add the absolute values:

$Area = \int_{0}^{π} sin x d x + \int_{π}^{2 π} sin x d x = 2 + 2 = 4$

Useful properties. $\int_{a}^{a} f (x) d x = 0$ , $\int_{b}^{a} f (x) d x = - \int_{a}^{b} f (x) d x$ , and an odd function integrated from $- a$ to $a$ gives 0.

Quick summary

The area under a velocity-time graph is the distance.
Estimate area with thin rectangles or trapeziums. Thinner strips are more accurate.
Trapezoidal rule: $\frac{h}{2} [y_{0} + 2 (middle heights) + y_{n}]$ .
Fundamental theorem: $\int_{a}^{b} f (x) d x = F (b) - F (a)$ .
Area below the axis gives a negative integral. For total area, split at the roots.

Quiz

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