The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = yᵃ is 364/3, is equal to :

Solution

Maths sir · 2 steps

The region lies between the y-axis and the curve, so integrate x = y^a with respect to y from 1 to 3.

1. Set up the strip. The bounds y = 1 and y = 3 are horizontal lines, so take thin horizontal strips. Each strip runs from x = 0 to x = yᵃ, and yᵃ > 0 on [1, 3], so its length is yᵃ.

   Area = ∫₁³ yᵃ dy = (3ᵃ⁺¹ - 1)/(a+1)

   a	(3ᵃ⁺¹-1)/(a+1)	Area
   3	(81 - 1)/4	20
   5	(729 - 1)/6	728/6 = 364/3
   7	(6561 - 1)/8	820
   9	(59049 - 1)/10	5904.8

2. So. Only a = 5 gives an area of 364/3, so the answer is (b).

The idea to keep. When the region is bounded by horizontal lines and a curve written as x in terms of y, integrate along y. Area is the integral of (right x minus left x) dy.

Answer from the NTA final answer key. Solution by PrepWiser.