Key Points
Key Points: Area Under the Curve
| Case | Formula |
|---|---|
| Area under y = f(x) | \[\int_a^bf(x)dx\] |
| Area between curves | \[\int_a^b[f(x)-g(x)]dx\] |
| Area w.r.t. y-axis x = g(y) |
\[\int_c^dg(y)dy\] |
| Even function | \[2\int_0^af(x)dx\] |
| Odd function | 0 |
If the area A lies below the X-axis, then A is negative, and in this case, we take | A |.
Key Points: Symmetry of Curve
| Type of Symmetry | Condition | Replacement Rule | Result |
|---|---|---|---|
| About the X–axis | (x, y) ∈ C ⇔ (x, -y) ∈ C | Replace y by -y | Curve is symmetric about the X–axis |
| About the Y–axis | (x, y) ∈ C ⇔ (-x, y) ∈ C | Replace x by -x | Curve is symmetric about the Y–axis |
| About Origin | Equation unchanged when both signs change | Replace x → -x, y → -y | The curve is symmetric about Origin |
Key Points: Standard Curves
Parabola:
-
y2 = 4ax → opens right
-
y2 = −4ax → opens left
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x2 = 4ay → opens upward
-
x2 = −4ay → opens downward
Ellipse:
-
\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\quad(a>b)\]
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\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\quad(a<b)\]
Important Questions [7]
- Solve the following: Find the area enclosed between the circle x^2 + y^2 = 1 and the line x + y = 1, lying in the first quadrant.
- Find the area of the region bounded by the parabola y2 = 16x and its latus rectum.
- Find the area of the region bounded by the curve y = x2 and the line y = 4.
- Find the area of the region bounded by the curve y = x2, and the lines x = 1, x = 2, and y = 0.
- Find the area of the region bounded by the curve y2 = 4x, the X-axis and the lines x = 1, x = 4 for y ≥ 0.
- Solve the following : Find the area of the region lying between the parabolas y^2 = 4x and x^2 = 4y.
- Find the area of the region lying between the parabolas y^2 = 4ax and x^2 = 4ay.
