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Find the area under the given curves an...

Find the area under the given curves and given lines:(i) `y=x^2,``x = 1, x = 2`and x-axis(ii) `y=x^4`, `x = 1, x = 5`and x-axis

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(i) Given curve `y=x^(2)` represents a parabola whose vertex is (0, 0).

` :. ` Required area (shaded region)
`int_(1)^(2)yd=int_(1)^(2)x^(2)dx`
`=[(x^(3))/(3)]_(1)^(2)=(1)/(3)(2^(3)-1^(3))`
` =(8)/(3)-(1)/(3)=(7)/(3)` sq. units.
(ii) Given , `y=x^(4)`
Here the degree of x is even, so the curve is symmetric about Y-axis and passes through the origin (0, 0).
` :. ` Required area= area between the curve `y=x^(4)`, X-axis, `x=1` and `x=5`.

`=int_(1)^(5)x^(5)dx=[(x^(5))/(5)]_(1)^(5)=(1)/(5)(5^(5)-1)`
`= (1)/(5)(3125-1)=(3124)/(5)` sq. units.
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Knowledge Check

  • The area bounded by the curve y^(2) = 14x and the lines x = 1, x = 4 above X-axis is

    A
    `7/3 sq. units`
    B
    `14/3 sq. units`
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    `28/3 sq. units`
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  • The area bounded by the curve y = x^(4) , X-axis and the lines x = 1, x = 5 is

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    `3124/5 sq. units `
    B
    `3126/5 sq. units`
    C
    `624/5 sq.units`
    D
    `626/5 sq. units`

  • The area bounded by the curve y = x^(2) , X-axis and the lines x = 1, x = 3 is

    A
    `26/3 sq. units`
    B
    `28/3 sq. units`
    C
    `1/3 sq. units`
    D
    9 sq. units

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