Find the area between y= x and f(x) = x^2

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

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.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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

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