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

Explanation

(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.