area of region bounded by curve and line in integration

Question

Find where the curve y=x^2-x-1 meets line y=5. Find area of the region bounded by the curve and line y=5

 

 

 

Answer 1

x^2 -x - 1 = 5

 

x^2 - x - 6 = 0

 

( x  - 3)(x  +  2 ) = 0

 

x=3 = 0 --> x=3 ---> (3,5)

 

x+2=0 ---> x= -2 ---> (-2, 5)

 

 

integral (5 - x^2 + x + 1)  = integral ( 6 + x - x^2)  = 6x + (1/2)x^2 - (1/3)x^3) ;  x=[-2,3]

 

X=3:  6(3) + (1/2)(3^2) - (1/3)(3^3) = 18 + 9/2 - 9 = 13.5

 

x=-2:  6(-2) + (1/2)(-2)^2 - (1/3)(-2)^3 = -12 + 2 - + 8/3 = -10 + 8/3 = -22/3

 

13.5 +22/3 = 20 and 5/6