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
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x2 = −4ay → opens downward
Ellipse:
-
\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\quad(a>b)\]
-
\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\quad(a<b)\]
Important Questions [10]
- Find the area of the region bounded by the parabola y2 = 25x and the line x = 5.
- Find the Area of the Ellipse X^2/4 + Y^2/25 = 1
- Area of the region bounded by y = x4, x = 1, x = 5 and the X-axis is _______.
- Find the area of the region bounded by y = x2, the X-axis and x = 1, x = 4.
- The slope of a tangent to the curve y = 3x2 – x + 1 at (1, 3) is ______.
- The area of the region bounded by the curve y = x2, x = 0, x = 3, and the X-axis is ______.
- Find the area between the two curves (parabolas) y2 = 7x and x2 = 7y.
- Find the Area of the Region Bounded by the Parabola Y2 = 16x and the Line X = 4.
- Find the area of the regions bounded by the line y = −2x, the X-axis and the lines x = −1 and x = 2.
- Find the area of the region bounded by the parabola y^2 = 4x and the line x = 3.
