# Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y.Then find the area of the region. 2y=5sqrt(x)  and 2y+2x=7

sciencesolve | Certified Educator

You should find first the limits of integration such that:

`{(2y=5sqrt x),(2y + 2x = 7):}`  => 2y + 8y^2/25 = 7 => 8y^2 + 50y - 175 = 0

`y_(1,2) = (-50+-sqrt(2500 + 5600))/16 => y_(1,2) = (-50+-sqrt8100)/16 => y_(1,2) = (-50+-90)/16`

`y_1 = 40/16 => y_1 = 10/4 => y_1 = 5/2`

`y_2 = -140/16 => y_2 = -70/4 => y_2 = -35/2`

You should evaluate the definite integral such that:

`int_(-35/2)^(5/2) (8y^2 + 50y - 175) dy`

Using the property of linearity of integral yields:

`int_(-35/2)^(5/2) (8y^2 + 50y - 175) dy = int_(-35/2)^(5/2) (8y^2)dy + int_(-35/2)^(5/2) (50y)dy - int_(-35/2)^(5/2) (175) dy`

`int_(-35/2)^(5/2) (8y^2 + 50y - 175) dy = (8y^3/3 + 50y^2/2 - 175y)|_(-35/2)^(5/2)`

`int_(-35/2)^(5/2) (8y^2 + 50y - 175) dy = (8y^3/3 + 25y^2 - 175y)|_(-35/2)^(5/2)`

`int_(-35/2)^(5/2) (8y^2 + 50y - 175) dy = 125/3 + 42875/3 + 625/4 - 765625/4 - 875/2 + 6125/2`

Hence, evaluating the definite integral, with respect to y, yields `int_(-35/2)^(5/2) (8y^2 + 50y - 175) dy = 125/3 + 42875/3 + 625/4 - 765625/4 - 875/2 + 6125/2.`