Find the area of the indicated region. We suggest you graph the curves to check whether one is above the other or whether they cross, and that you use technology to check your answer. HINT [See Example 3.]
Enclosed by y = x and y = x^4
look at the solution I just gave catlyn in the previous post
compare your graph with hers
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E4+%2C+y+%3D+x
neat eh?
To find the area of the region enclosed by the curves y = x and y = x^4, we need to calculate the definite integral of the difference between the two functions.
Step 1: Graph the curves
First, let's graph the curves y = x and y = x^4 to visualize the region we are interested in. You can use any graphing tool or software to plot the curves.
Step 2: Determine the interval of integration
Next, we need to determine the interval over which we are calculating the area. To do this, we find the x-values where the curves intersect. Set the two equations equal to each other and solve for x:
x = x^4
Rearrange the equation:
x^4 - x = 0
Factor out an x:
x(x^3 - 1) = 0
Set each factor equal to zero:
x = 0, x^3 - 1 = 0
Solving for x^3 - 1 = 0, we get:
x^3 = 1
Taking the cube root of both sides, we get:
x = 1
Therefore, the interval of integration is from x = 0 to x = 1.
Step 3: Set up the definite integral
Now that we have the interval, we can set up the definite integral to calculate the area between the curves:
A = ∫[0,1] (x^4 - x) dx
Step 4: Evaluate the integral
To find the area, we need to evaluate the integral. You can either calculate it by hand or use technology such as a graphing calculator or software.
After evaluating the integral, you will have the area of the region enclosed by the curves y = x and y = x^4. Remember to check your answer using technology to ensure accuracy.