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 = x4
its is x^4 if anyone want to help
The curves intersect at (0,0) and (1,1)
so the area is just
∫[0,1] (x - x^4) dx = ____
Can you give me a definitive answer? For some reason I can't quite get the answer from that function
really? Just apply the power rule
∫x^n dx = 1/(n+1) x^(n+1)
so,
∫[0,1] (x - x^4) dx = 1/2 x^2 - 1/5 x^3 [0,1] = (1/2 - 1/5) - (0 - 0) = 3/10
next time you get stuck, post your work, so we can see what went wrong.
To find the area of the region enclosed by the two curves, we need to determine the points of intersection first. Let's set the two equations equal to each other and solve for x:
x = x^4
Now, we can solve this equation by rearranging it:
0 = x^4 - x
Next, we factor out an x:
0 = x(x^3 - 1)
Setting each factor equal to zero gives us:
x = 0 (for x)
x^3 - 1 = 0
Solving for x^3:
x^3 = 1
And taking the cube root of both sides:
x = 1
Now that we have the points of intersection (x = 0 and x = 1), we can graph the curves to see which one is above the other.
Graphing y = x and y = x^4 on a coordinate plane, we can see that the curve y = x^4 is above y = x between the points x = 0 and x = 1.
Now, to find the area, we integrate the function that is on top (in this case, y = x^4) and subtract the integral of the function that is on the bottom (y = x).
Therefore, the area (A) of the region enclosed by y = x and y = x^4 is given by:
A = ∫[0, 1] (x^4 - x) dx
Integrating this from x = 0 to x = 1 will give us the area of the region. You can use technology or software, such as a graphing calculator or computer algebra system, to evaluate this integral and find the exact value of the area.