Find the value of m such that y=mx divides the area under the curve y=(1-x)x in [0,1] into two regions of equal area

D: D: D:

To find the value of m such that the line y = mx divides the area under the curve y = (1-x)x in [0,1] into two regions of equal area, we need to set up an equation and solve for m.

Let's start by finding the equation of the line y = mx, where m is the slope.

Now, to find the area under the curve y = (1-x)x in [0,1], we need to integrate the equation over the interval [0,1]. We can use definite integration to achieve this.

The integral of y = (1-x)x with respect to x over the interval [0,1] is given by:

A = ∫[0,1] (1-x)x dx

To divide this area into two equal regions, we need to set up the equation:

A1 = A2

Where A1 is the area under the curve y = (1-x)x above the line y = mx, and A2 is the area under the curve below the line.

Now, let's find the individual areas A1 and A2.

A1 = ∫[0, p] (1-x)x dx
A2 = ∫[p, 1] (1-x)x dx

where p is the point of intersection between the curve and the line.

We can set up the integral and solve it using calculus, but it might be easier to use some symmetry properties of the curve to simplify the equation.

The curve y = (1-x)x is symmetric around x = 1/2. This means that the area of A1 and A2 will be equal when the line y = mx intersects the curve at x = 1/2.

To find m, we substitute x = 1/2 into the equation of the line y = mx:

y = m(1/2)
m/2 = (1/2)m

Now, to find the value of m, we can cancel out the common factor of 1/2:

m/2 = m

Multiplying through by 2:

m = 2m

Dividing both sides by m:

1 = 2

Uh oh! This equation is a contradiction. It means that there is no value of m that will divide the area under the curve y = (1-x)x in [0,1] into two regions of equal area.

Therefore, there is no value of m that satisfies the given condition.

you want m such that

∫[0,1] (x-x^2)-mx dx = ∫[0,1] mx dx
∫[0,1] x-x^2-2mx dx = 0
(1-2m)/2 x^2 - 1/3 x^3 [0,1] = 0
1 - 2m - 1/3 = 0
m = 1/3