CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Q335 Marksparabola_area
CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Question 33 (Applications Of Integrals)
Using integration, find the area enclosed by \(y=x^2\) and \(y=2\). OR Using integration, find the area inside \(9x^2+16y^2=144\) between \(x=-2\) and \(x=2\).
Alternative 1 5 Marks
Using integration, find the area enclosed by \(y=x^2\) and \(y=2\).
Answer: 8√2/3 square units.
- The intersections have x=±√2. The line lies above the parabola between them.
- By symmetry the area is 2∫₀^√2(2−x²)dx=2[2x−x³/3]₀^√2.
- The result is 8√2/3 square units.
Alternative 2 5 Marks
Using integration, find the area inside \(9x^2+16y^2=144\) between \(x=-2\) and \(x=2\).
Answer: 6√3+4π square units.
- The upper ellipse branch is 3√(1−x²/16). Use both vertical halves and symmetry about the y-axis.
- The area is 12∫₀²√(1−x²/16)dx. Put x=4 sin t, giving 48∫₀^(π/6)cos²t dt.
- Evaluate 24[t+sin(2t)/2]₀^(π/6)=4π+6√3.