Formulae [2]
Formula: Modulus Functions
Break modulus into cases:
\[|x-a|=
\begin{cases}
x-a, & x\geq a \\
a-x, & x<a & &
\end{cases}\]
Formula: Area between Two Curves
\[\text{Area between two curves}=\int_a^b[\text{upper curve - lower curve}]dx\]
\[\text{Area between two curves}=\int_{a}^{b}y\mathrm{of}f(x)dx-\int_{a}^{b}y\mathrm{of}g(x)dx\]
Key Points
Key Points: Geometrical Interpretation of Definite Integral
The area bounded by the curve y = f (x), the x-axis and the ordinates. x = a, x=b is \[\int_a^bydx\].
Sign of Area:
| Condition | Result |
|---|---|
| Curve above the x-axis | Area is positive |
| Curve below the x-axis | Area is negative |
| Curve cuts the x-axis | Integral ≠ actual area |
Key Points: Symmetry
| Type of Symmetry | What to Replace | Condition | Result |
|---|---|---|---|
| About y-axis | Replace (x) by (-x) | Equation unchanged | Symmetrical about the y-axis |
| About x-axis | Replace (y) by (-y) | Equation unchanged | Symmetrical about the x-axis |
| About origin | Replace (x) by (-x), (y) by (-y) | Equation unchanged | Symmetrical about the origin |
| About y = x | Interchange (x) and (y) | Equation unchanged | Symmetrical about the line y = x |
| About y = −x | Replace (x) by (-y), (y) by (-x) | Equation unchanged | Symmetrical about the line y = −x |
Key Points: Area Under a Curve
-
If the curve is ,
\[\mathrm{Area}=\int_a^bf(x)dx\] -
If curve is x = g(y),
\[\mathrm{Area}=\int_c^dg(y)dy\] -
If the curve is on both sides → split + add
When to Use:
| Curve form | Formula |
|---|---|
| y = f(x) | \[\int ydx\] |
| x = f(y) | \[\int xdy\] |
Key Points: Standard Curves
| Curve | Shape |
|---|---|
| \[y=\sqrt{a^2-x^2}\] | Upper semicircle |
| \[x^2+y^2=a^2\] | Circle |
| \[y^2=4ax\] | Right parabola |
| \[x^2=4ay\] | Upward parabola |
| y = sin x,cos x | Wave (sign changes!) |
