Finding Area Between Polynomial Graphs

Find the indefinite integral of the polynomial f(x) = 6x^2 - 4x.

2x^3 - 2x^2 + C

Find the area enclosed between the curve y = x^2 and the x-axis between x = 0 and x = 3.

9 (Integral of x^2 dx from 0 to 3 is [x^3 / 3], which gives 27/3 - 0 = 9).

Find the intersection points of the line y = 4 and the parabola y = x^2.

x = -2 and x = 2 (Solve x^2 = 4).

Calculate the area enclosed between y = 4 and y = x^2.

32/3 or 10.67 (Integrate 4 - x^2 from -2 to 2: [4x - x^3 / 3] gives (8 - 8/3) - (-8 + 8/3) = 32/3).

Find the intersection points of y = 3x and y = x^2.

x = 0 and x = 3 (Solve x^2 - 3x = 0, giving x(x - 3) = 0).

Calculate the area enclosed between the curves y = 3x and y = x^2.

4.5 or 9/2 (Integrate 3x - x^2 from 0 to 3: [1.5x^2 - x^3 / 3] gives 13.5 - 9 = 4.5).

Calculate the area enclosed between the line y = x + 2 and the parabola y = x^2.

4.5 or 9/2 (Intersections at x = -1 and x = 2; integrate (x + 2 - x^2) from -1 to 2).

Calculate the area enclosed between the two parabolas y = 2 - x^2 and y = x^2.

8/3 or approximately 2.67 (Intersections at x = -1 and x = 1; integrate (2 - 2x^2) from -1 to 1: [2x - 2x^3 / 3] gives 4/3 - (-4/3) = 8/3).