`y_1 = x^2 + 2x +1 , y_2 = 2x + 5` Set up the definite integral that gives the area of the region

Given the curve equations ,they are

`y_1=x^2 + 2x +1` -----(1)

`y_2=2x + 5` -----(2)

to get the boundaries or the intersecting points of the curves we have to equate the functions .

y_1=y_2

=> `x^2 + 2x +1= 2x + 5`

=> `x^2-4 =0`

=> `(x+2)(x-2)=0`

=> `x=2 or x=-2`

So,

The area = `int _(-2) ^2 ((2x + 5) -(x^2 + 2x +1)) dx `

= `int _(-2) ^2 (4-x^2) dx `

= `[4x - x^3 /3]_(-2) ^2 `

=`[8-8/3] -[-8 +8/3] = 8-8/3 +8 -8/3 =16 -16/3 = 16(1-1/3) = 16*2/3 = 32/3 `

so area of the region enclosed by the curves is = `32/3`

Comments

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