Find the area of the region between y=x1/2 and y=x1/4 for 0≤x≤1.
To find the area of the region between the two curves, we need to find the points of intersection first.
x^(1/2) = x^(1/4)
Squaring both sides, we get:
x = x^(1/2)^2 = x^(1/4)^4 = x^(1/2)
Subtracting x^(1/2) from both sides, we get:
0 = x^(1/2) - x
Squaring again, we get:
x = x - 2x^(3/2) + x^2
Simplifying, we get:
x^(3/2) - x^2 = 0
Factoring out x^(3/2), we get:
x^(3/2)(1 - x^(1/2)) = 0
Therefore, x = 0 or x = 1.
Now we can find the area of the region using integration:
∫(x^(1/2) - x^(1/4)) dx from x=0 to x=1
= [2/3x^(3/2) - 4/5x^(5/4)] from x=0 to x=1
= 2/3 - 4/5
= 2/15
Therefore, the area of the region between y=x^(1/2) and y=x^(1/4) for 0≤x≤1 is 2/15 square units.
To find the area of the region between the curves y = x^(1/2) and y = x^(1/4) for 0≤x≤1, we need to find the definite integral of the difference of the two curves over the given interval.
First, let's set up the integral:
∫[0,1] (x^(1/2) - x^(1/4)) dx
To solve this integral, we need to integrate each term separately.
Let's first integrate x^(1/2):
∫[0,1] x^(1/2) dx
Using the power rule for integration, we add 1 to the exponent and divide by the new exponent:
= (2/3)x^(3/2) evaluated from 0 to 1
= (2/3)(1^(3/2) - 0^(3/2))
= (2/3)(1 - 0)
= 2/3
Now let's integrate x^(1/4):
∫[0,1] x^(1/4) dx
Using the power rule for integration, we add 1 to the exponent and divide by the new exponent:
= (4/5)x^(5/4) evaluated from 0 to 1
= (4/5)(1^(5/4) - 0^(5/4))
= (4/5)(1 - 0)
= 4/5
Now we can find the area between the two curves by subtracting the integral of x^(1/4) from the integral of x^(1/2):
(2/3) - (4/5)
= (10/15) - (12/15)
= -2/15
Since the area cannot be negative, we take the absolute value:
| -2/15 |
= 2/15
Therefore, the area of the region between y = x^(1/2) and y = x^(1/4) for 0≤x≤1 is 2/15.
To find the area of the region between two curves, we need to determine the points of intersection.
The given equations are y = x^(1/2) and y = x^(1/4).
To find the points of intersection, we need to solve the equation x^(1/2) = x^(1/4).
Let's solve for x:
x^(1/2) = x^(1/4)
Squaring both sides:
(x^(1/2))^2 = (x^(1/4))^2
x = x^(1/2 + 1/2)
x = x^(1)
To find the points of intersection, we equate the two expressions for x:
x = x^(1)
x - x^(1) = 0
x(1 - x^(1-1)) = 0
x(1 - 1) = 0
x = 0
So, one point of intersection is x = 0.
Now, let's find the second point of intersection:
Since x = 0 is already a point of intersection, we consider the other possibility x^(1) = 1.
This means x = 1.
So, the two points of intersection are x = 0 and x = 1.
To find the area between the curves, we integrate the difference of the two curves with respect to x within the given limits of integration.
The area (A) can be calculated using the following formula:
A = ∫[0,1] (x^(1/4) - x^(1/2)) dx
Evaluating this integral will give us the area between the curves.