Application of Integrals · Question 25

JEE Main 2025 (Online) 3rd April Morning ShiftNumericalMedium📈 Topic in 6% papers+4

The area of the region bounded by the curve y = \|x|, x|x-2|\, the x-axis and the lines x = -2 and x = 4 is equal to

Answer

For -2 x < 0: x|x-2| < 0 < |x| y = -x

For 0 x 2: x(2-x) x x 1

y = 2x - x2 on [0, 1], y = x on [1, 2]

For 2 x 4: x(x-2) x x 3

y = x on [2, 3], y = x2 - 2x on [3, 4]

[figure]

Area = -20(-x) dx + 01(2x - x2) dx + 13x dx + 34(x2 - 2x) dx

= 2 + (1 - 13 ) + 9-12 + [ x33 - x2 ]34

= 2 + 23 + 4 + ( 163 - 0 )

= 12

Answer: 12

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