Application of Integrals
Definite integral ka sabse khoobsurat istemaal hai — kisi curve ke neeche ka area nikaalna. Ye chapter dikhata hai ki integration se areas kaise calculate karte hain.
1. Area Under a Curve
Curve \(y = f(x)\), \(x\)-axis, aur lines \(x = a\) tatha \(x = b\) ke beech ka area: \[ A = \int_{a}^{b} y\,dx = \int_{a}^{b} f(x)\,dx, \qquad f(x) \ge 0. \]
\(y = x^{2}\), \(x = 0\) se \(x = 3\) tak \(x\)-axis ke saath area: \[ A = \int_{0}^{3} x^{2}\,dx = \left[\frac{x^{3}}{3}\right]_{0}^{3} = 9 \ \text{sq units}. \]
2. Area with respect to \(y\)-axis
Agar curve \(x = g(y)\) ho aur area \(y = c\) se \(y = d\) tak chahiye:
\[ A = \int_{c}^{d} x\,dy = \int_{c}^{d} g(y)\,dy. \]
3. Area Below the \(x\)-axis
Jahan curve \(x\)-axis ke neeche ho, wahan integral negative aata hai. Area hamesha positive hota hai, isliye absolute value lo:
\[ A = \left| \int_{a}^{b} f(x)\,dx \right|. \]
Agar curve kuch jagah upar aur kuch neeche ho, to intervals alag-alag karke har part ka area positive lo, phir jodo.
4. Area Between Two Curves
Do curves \(y = f(x)\) (upar) aur \(y = g(x)\) (neeche) ke beech ka area, \(x = a\) se \(x = b\) tak: \[ A = \int_{a}^{b} \big[\,f(x) - g(x)\,\big]\,dx. \]
Pehle intersection points nikaalo (\(f(x) = g(x)\)) — wahi limits \(a\) aur \(b\) dete hain.
\(y = x\) aur \(y = x^{2}\) ke beech ka area. Intersection: \(x = x^{2} \Rightarrow x = 0, 1\). Yahan \(x \ge x^{2}\) (0 se 1 ke beech), toh \[ A = \int_{0}^{1} (x - x^{2})\,dx = \left[\frac{x^{2}}{2} - \frac{x^{3}}{3}\right]_{0}^{1} = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}. \]
5. Area of Standard Shapes (verification)
Integration se classical formulas bhi nikalte hain — jaise circle \(x^{2} + y^{2} = r^{2}\) ka area integrate karke \(\pi r^{2}\) aata hai.
Key Takeaways
- Curve ke neeche area \(= \int_a^b f(x)\,dx\) (jab \(f \ge 0\)).
- \(x\)-axis ke neeche wale part me absolute value lo.
- Do curves ke beech: \(\int_a^b [\text{upar} - \text{neeche}]\,dx\); pehle intersection points nikaalo.
- \(y\)-axis ke respect me area ke liye \(\int x\,dy\) use karo.