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Application of Integrals – Area of Circle & Ellipse
Using definite integrals to derive area formulas for standard curves (circle and ellipse).
Idea: integrate the upper half of the curve and double it, using symmetry about the x-axis.
1. Area of a Circle using Integrals
Equation of the circle
Circle of radius \(r\):\quad \(x^2 + y^2 = r^2\)
Upper semicircle: \(\;y = \sqrt{r^2 - x^2}\)
Area by integration
- Use symmetry about the x-axis.
- Integrate the upper semicircle from \(-r\) to \(+r\) and multiply by 2.
\[
A = 2\int_{-r}^{r} \sqrt{r^2 - x^2}\,dx
\]
Use substitution \(x = r\sin\theta\). After simplification,
\[
A = \pi r^2
\]
2. Area of an Ellipse using Integrals
Equation of the ellipse
Standard ellipse with semi-axes \(a\) (x-direction) and \(b\) (y-direction):
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Upper half:
\[
y = b\sqrt{1 - \frac{x^2}{a^2}}
\]
Area by integration
- Again, use symmetry about the x-axis.
- Integrate the upper half from \(-a\) to \(+a\) and double it.
\[
A = 2\int_{-a}^{a} b\sqrt{1 - \frac{x^2}{a^2}}\,dx
\]
Substitute \(x = a\sin\theta\). Then \(dx = a\cos\theta\,d\theta\), and limits change from
\(-a \to -\frac{\pi}{2}\) to \(+a \to \frac{\pi}{2}\).
\[
A = 2b\int_{-\pi/2}^{\pi/2} b\sqrt{1-\sin^2\theta}\,a\cos\theta\,d\theta
= 2ab\int_{-\pi/2}^{\pi/2} \cos^2\theta\,d\theta
= \pi ab
\]
3. Comparison – Circle vs Ellipse
| Shape |
Equation |
Integral Expression |
Final Area |
| Circle |
\(x^2 + y^2 = r^2\) |
\(2\displaystyle\int_{-r}^{r} \sqrt{r^2 - x^2}\,dx\) |
\(\pi r^2\) |
| Ellipse |
\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) |
\(2\displaystyle\int_{-a}^{a} b\sqrt{1 - x^2/a^2}\,dx\) |
\(\pi ab\) |
Note: A circle is a special ellipse with \(a = b = r\). Then \(\pi ab \Rightarrow \pi r^2\).
4. Quick Concept Insight
- Both derivations rely on symmetry and definite integrals.
- Circle: single parameter \(r\) (radius).
- Ellipse: two parameters \(a\) and \(b\) (semi-major and semi-minor axes).
- Formula pattern: \(\text{Area} = \pi \times (\text{semi-axis in x}) \times (\text{semi-axis in y})\).
Application of Integrals
Area under Curves
Circle & Ellipse