find the definite integral that represents the are of the region enclosed by y= 4x-x^2.
To find the definite integral that represents the area of the region enclosed by the curve y = 4x - x^2, you can follow these steps:
1. Start by determining the x-values where the curve intersects with the x-axis. These are the points where y = 0.
To find these points, set 4x - x^2 = 0 and solve for x:
0 = 4x - x^2
Rearrange the equation:
x^2 - 4x = 0
Factor out x:
x(x - 4) = 0
Solve each factor:
x = 0 or x - 4 = 0
x = 0 or x = 4
So, the curve intersects the x-axis at x = 0 and x = 4.
2. Next, determine the limits of integration for the definite integral. The limits should be the x-values of the points where the curve intersects the x-axis.
The lower limit of integration (a) is the smaller x-value, which in this case is 0.
The upper limit of integration (b) is the larger x-value, which is 4.
Therefore, the limits of integration for the definite integral are from 0 to 4: ∫[0, 4].
3. Now, write the integrand for the definite integral. The integrand represents the area between the curve and the x-axis, which is given by the absolute value of the function.
In this case, the integrand is |4x - x^2|.
4. Finally, set up and solve the definite integral:
∫[0, 4] |4x - x^2| dx
To solve the integral, you can split it into two separate integrals, one from 0 to 2 and the other from 2 to 4, since the function changes direction at x = 2.
∫[0, 2] (4x - x^2) dx + ∫[2, 4] (x^2 - 4x) dx
Evaluate each integral separately, and then add the results together to get the final area enclosed by the curve.