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Area of a Region Bounded by a Curve and a Line

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Area of a Region Bounded by a Curve and a Line - Lesson Summary

Equation of the parabola is y2 = 6x and equation of the line is y = 4.

The entire shaded region if we move it horizontally from y =0 to y = 4.

Area of the region = ∫y=ay=b f(y) dy

Height of strip = Function of parabola - Function of Y - axis = y2/6 - x

                       = y2/6 - 0 = y2/6

                      ∴ y2/6 = f(y)

Area of the region = ∫y=0y=4 f(y) dy

                          = ∫y=0y=4 y2/6 dy

                          = 1/6 .∫y=0y=4 y2 dy = 1/6[y3/3]04

                         = 1/6[43/3  - 0/3]

                         = 1/6[43/3] = 64 / 18 = 32 / 9 sq.units

Therefore, the area of the required region is 32/9 square units.

Alternatively, the area of the shaded region can also be found by considering a strip of width dx parallel to the y-axis.

Area of the region = ∫x=ax=b f(x) dx

Height of strip = Function of line - Function of parabola

f(x) = 4 - √(6x)

Area of the region = ∫x=ax=b (4 - √(6x)) dx

Solving y2 = 6x and y = 4, we get

16 = 6x        ⇒ x = 16/6 = 8/3          ∴ A = (8/3 , 4)

Area of the region = ∫08/3 (4 - √(6x)) dx

                           = [4x - √6 . x3/2/(3/2)]08/3

                           = [4 x 8/3 - 2/3 √6 . (8/3)3/2] - 0

                           = 32/3 - 64/9 = 32/9 sq. Units.

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