Given the line y= x and the parabola f(x)= x^2.
To find the area between the line and the parabola, we need to determine the intersection points between the line and the curve.
To find the intersection points, we will caculate x values such that y = f(x).
==> x^2 = x
==> x^2 - x = 0
==> x ( x-1) = 0
==> x= 0 , 1
Now we will determine the area under the line y= x between x= 0 and x= 1.
==> A1 = integral y
= intg x dx
= x^ 2/2
==> A1 = (1/2) - 0/2 = 1/2
Now we will calculate the area under the curve f(x) = x^2 and x= 0 and x= 1.
==> A2 = intg f(x) dx
= intg x^2 dx
= x^3 /3 + C
==> A2 = 1/3 - 0 = 1/3
Then, the area between y and f(x) is:
A = A1 - A2 = 1/2 - 1/3 = 1/6
Then the area = 1/6 square units.
You've given only the limit curves but you did not gave the limit lines.
We'll calculate the area located between the 2 given curves, using Leibniz-Newton formula:
Int (x^2 - x)dx = F(b) - F(a), where x = a and x = b are the limit lines.
We'll determine the indefinite integral first:
Int (x^2 - x)dx = Int x^2 dx - Int x dx
Int (x^2 - x)dx = x^3/3 - x^2/2
F(b) = b^3/3 - b^2/2
F(a) = a^3/3 - a^2/2
Int (x^2 - x)dx = b^3/3 - b^2/2 - a^3/3 + a^2/2