Find the area of the shaded region.
The area is shaded from 0,2 on the x-axis
Given: y=0 and y= x^4-2x^3
This is just a straightforward integration of powers of x. What answer do you get? Or, where do you get stuck?
To find the area of the shaded region, we need to determine the interval over which the shaded region lies and then integrate the function that defines the region's boundary.
First, let's analyze the given functions:
- The x-axis limits the shaded region from 0 to 2 on the x-axis.
- The boundary of the shaded region is defined by the functions y = 0 and y = x^4 - 2x^3.
To find the points where the boundary curves intersect, we set the two functions equal to each other:
0 = x^4 - 2x^3
Next, we solve for x:
x^4 - 2x^3 = 0
x^3(x - 2) = 0
From this, we find two solutions:
x = 0 and x = 2.
We now know that the shaded region occupies the interval from x = 0 to x = 2.
To find the area, we integrate the difference between the two functions over this interval:
Area = ∫[0, 2] (x^4 - 2x^3) dx
Simplifying the integral, we get:
Area = ∫[0, 2] (x^4 - 2x^3) dx
= ∫[0, 2] (x^4) dx - ∫[0, 2] (2x^3) dx
Integrating each term separately:
Area = (1/5)x^5 - (1/2)x^4 | [0, 2] - (1/2)x^4 | [0, 2]
Now, we can substitute the upper and lower limits of integration:
Area = (1/5)(2^5) - (1/2)(2^4) - (1/2)(0^4) + (1/2)(0^4)
= (1/5)(32) - (1/2)(16)
= 6.4 - 8
= -1.6
So, the area of the shaded region is -1.6 square units.