Find the area of the shaded region.
y= 0
y= x^4-2x^3
To find the area of the shaded region, we first need to determine the points of intersection between the two curves, y = 0 and y = x^4 - 2x^3.
Setting the two equations equal to each other, we get:
0 = x^4 - 2x^3
Next, we can factor out an x^3 from the right side of the equation:
0 = x^3(x - 2)
Setting each factor equal to zero gives us two equations to solve:
x^3 = 0 (equation 1)
x - 2 = 0 (equation 2)
Solving equation 1 for x gives us x = 0. Plugging this value into equation 2 gives us x = 2.
Therefore, the two points of intersection are (0, 0) and (2, 0).
Now, we can integrate to find the area between the curves. Since the region is bounded by y = 0 and y = x^4 - 2x^3, the integral representing the area is:
Area = ∫[from 0 to 2] (x^4 - 2x^3) dx
Evaluating this integral gives us the area of the shaded region.