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: Area Under Simple Curves
| Case | Standard Form | Area Formula |
| Region above x-axis | y = f(x) | \[A = \int_{a}^{b} y \, dx\] |
| Region bounded by y-axis | x = g(y) | \[A = \int_{c}^{d} x \, dy\] |
| Curve below x-axis | y = f(x) < 0 | \[A = \left\vert \int_{a}^{b} f(x) \, dx \right\vert\] |
| Curve crossing x-axis | Mixed signs | \[A = \vert A_1 \vert + A_2\] |
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: 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: 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!) |
Important Questions [10]
- Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x^2 + y^2 = 32.
- Using integration find the area of the region {(x, y) : x2+y2⩽ 2ax, y2⩾ ax, x, y ⩾ 0}.
- Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle
- Using the Method of Integration, Find the Area of the Triangle Abc, Coordinates of Whose Vertices Are a (4 , 1), B (6, 6) and C (8, 4).
- Find the Area Enclosed Between the Parabola 4y = 3x2 and the Straight Line 3x - 2y + 12 = 0.
- Find the Area Bounded by the Circle X2 + Y2 = 16 and the Line `Squareroot 3 Y = X` in the First Quadrant, Using Integration.
- Using Integration, Find the Area of the Region {(X, Y) : X2 + Y2 ≤ 1 ≤ X + Y}.
- Find the Area of the Smaller Region Bounded by the Ellipse X 2 9 + Y 2 4 = 1 and the Line X 3 + Y 2 = 1 .
- Find the Area of the Region. {(X,Y) : 0 ≤ Y ≤ X2 , 0 ≤ Y ≤ X + 2 ,-1 ≤ X ≤ 3} .
- Using Integration Find the Area of the Triangle Formed by Negative X-axis and Tangent and Normal to the Circle X 2 + Y 2 = 9 at ( − 1 , 2 √ 2 ) .
