Class 12 Applications Of Integrals · Short Answer2 Marksarea_by_integration
Class 12 Maths · Applications Of Integrals · Short Answer #10
Sketch the region between y=√(4−x²) and the x-axis, and find its area by integration.
✅ Step-by-Step Solution
Final Answer: \(2 \pi\) square units
- The curve is the upper semicircle x²+y²=4, with x-axis endpoints (−2,0),(2,0). The strip widths are \(\sqrt{4 - x^{2}}\) for x∈\(\left[-2, 2\right]\).
- Primitive \(\frac{x \sqrt{4 - x^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{x}{2} \right)}\), endpoint difference \(2 \pi\).
- Therefore the area is \(2 \pi\) square units.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The curve is the upper semicircle x²+y²=4, with x-axis endpoints (−2,0),(2,0). The strip widths are \(\sqrt{4 - x^{2}}\) for x∈\(\left[-2, 2\right]\).
- [1 mark] Primitive \(\frac{x \sqrt{4 - x^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{x}{2} \right)}\), endpoint difference \(2 \pi\).
- [0.5 marks] Therefore the area is \(2 \pi\) square units.
Master Chapter 8: Applications Of Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Applications Of Integrals.
← Previous Question
Find the area bounded by y=x², the x-axis, and x=−1,x=1.
Next Question →For a>0, find the area under y=√(a²−x²), above the x-axis, between…
More Practice Questions in Applications Of Integrals
Related Questions in Applications Of Integrals
Class 12 Applications Of Integrals · Assertion Reason
Assertion: The area between y=sin²x and the x-axis on [0,π] is greater than that for y=sin x…
Class 12 Applications Of Integrals · Assertion Reason
Assertion: The area inside 2x²+3y²=6 is greater than the area inside x²+y²−2x+4y+4=0. Reason…
Class 12 Applications Of Integrals · Assertion Reason
Assertion: The area enclosed by x²+y²=36 equals 36π square units. Reason: A circle x²+y²=r²…
Class 12 Applications Of Integrals · Assertion Reason
Assertion: The first-quadrant region enclosed by y=x³ and y=x has area 1/4 square unit…
Class 12 Applications Of Integrals · Assertion Reason
Assertion: The region between x=y² and x=3 has area 4√3 square units. Reason: Intersections…
Class 12 Applications Of Integrals · Case Study
An external elevator has an upper-semicircular floor. Its curved boundary is the upper half…