Sketch the region enclosed by x + (y^2) =56  and  x+y=0. Decide whether to integrate with respect to x or y. Then find the area of the region.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the limits of integration, hence, you need to solve for x and y the following system of equations such that:

`{(x + y^2 = 56),(x + y = 0):}` => `{(y^2 - y - 56 = 0),(x = -y):}`

You should use quadratic formula to solve the first equation such that:

`y_(1,2) = (1 +- sqrt(1 + 224))/2 => y_(1,2) = (1 +- sqrt225)/2`

`y_(1,2) = (1 +- 15)/2 => y_1 = 8 ; y_2 = -7`

`y_1 = 8 => x_1 = -8`

`y_2 = -7 => x_2 = 7`

You need to evaluate the definite integral such that:

`int_(-7)^8 (56 - y^2 + y) dy = int_(-7)^8 56 dy - int_(-7)^8 y^2 dy + int_(-7)^8 y dy`

`int_(-7)^8 (56 - y^2 + y) dy = (56y - y^3/3 + y^2/2)|_(-7)^8`

`int_(-7)^8 (56 - y^2 + y) dy = 56(8 + 7) - 8^3/3 - 7^3/3 + 64/2 - 49/2`

`int_(-7)^8 (56 - y^2 + y) dy = 840 - 855/3 + 15/2`

`int_(-7)^8 (56 - y^2 + y) dy = (5040 - 1710 + 45)/6`

`int_(-7)^8 (56 - y^2 + y) dy = 3375/6`

`int_(-7)^8 (56 - y^2 + y) dy = 562.5`

Hence, evaluating the area of the region, integrating with respect to y, yields `int_(-7)^8 (56 - y^2 + y) dy = 562.5.`

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