Calculate the area of the figure bounded by the parabola y = 4x-x ^ 2 and the straight line

Calculate the area of the figure bounded by the parabola y = 4x-x ^ 2 and the straight line passing through the points A (4; 0) and B (0; 4).

Let us find the equation of the straight line, for this we solve the system of equations:

4 * k + b = 0 and b = 4,

4 * k = -b,

4 * k = -4,

k = -1.

Therefore, the equation of the straight line has the form:

y = -x + 4 = 4 – x.

Find the intersection points of the graphs of the straight line and the parabola, solve the equation:

4 * x – x² = 4 – x,

-x² + 5 * x – 4 = 0.

Equation roots:

x = 4; y = 4 – x = 0,

x = 1; y = 4 – x = 3.

Therefore, the limits of integration will be as follows:

x = 1 and x = 4.

The required area is the difference:

s = integral (1, 4) (4 * x – x²) dx – area of a triangle with vertices (1; 0), (4; 0), (1; 3).

s = 9 – (1/2) * 3 * 3 = 9 – 9/2 = 9/2 units ².