How do i calculate the area by integral

y=x^2+1 and y=2-x^2

first find the points of intersection :

x^2 + 1 = 2-x^2

2 x^2 = 1

=> x = +/- 1/sqrt(2)

so
.....1/sqrt(2)
A = ∫ [2 - x^2 - (x^2 + 1)] dx
...-1/sqrt(2)

..= [x - 2/3 x^3 ] from -1/sqrt(2) to 1/sqrt(2)

...= 2/sqrt(2) - 2/3 ( [1/sqrt(2)]^3 - [-1/sqrt(2)]^3 )

...= sqrt(2) - 2/3 [2 (1/2sqrt(2))]

...= sqrt(2) - 2 /3sqrt(2)

∫ ( x² + 1 )dx = ⅓x³ + x + C₀

C₀ is a constant of integration since this is an indefinite integral. The rules for carrying out integrations by polynomials are pretty straight forward. You can try the next one on your own.

just integral it !and you must know the range of X
1-integral y = x^3/3 + x+ c
2-integral y = 2x-(x^3/3)