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JEE Main 202226 JulyMorning Shift
Exam
JEE Main
Year
2022
Date
26 July
Shift
Morning Shift
Subject
Mathematics
Chapter
Area Under Curves

JEE Main 2022 · 26 July · Morning Shift · Mathematics · Area Under Curves

The odd natural number a, such that the area of the region bounded...

The odd natural number a, such that the area of the region bounded by y=1,y=3,x=0,x=yay = 1, y = 3, x = 0, x = {y}^{a} is 3643\dfrac{364}{3}, equal to:The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = y ^ a is (364)/(3) ,

Options

  1. A33
  2. B55
  3. C77
  4. D99

Correct answer

B. 55

Step-by-step solution

Plotting the graph of the given functions, we get the shaded region as the area bounded by the curves.

The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = y ^ a is (364)/(3) ,

Given, area of shaded region =3643= \dfrac{364}{3}

⇒∫13yady=3643⇒ {∫}_{1}^{3}{y}^{a}dy = \dfrac{364}{3}

⇒[ya+1a+1]13=3643⇒ {\left[\dfrac{{y}^{a + 1}}{a + 1}\right]}_{1}^{3} = \dfrac{364}{3}

⇒3a+1−1a+1=3643=729−16=36−16⇒ \dfrac{{3}^{a + 1} - 1}{a + 1} = \dfrac{364}{3} = \dfrac{729 - 1}{6} = \dfrac{{3}^{6} - 1}{6}

⇒a=5⇒ a = 5

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