Find the area of the region bounded by the curves y=x^(-1/2), y=x^(-2), y=1 and y=3.
You get:
a.) 1/2(sqrt(3)) + 4/3
b.) 2(sqrt(3)) - 8/3
c.) 1/2(sqrt(3) - 32/3
d.) 2(sqrt(3)) - 32/3
e.) 8/3 - 2(sqrt(3))
The definite integral of y^-.5-y^-2 evaluated from 1 to 3. I forgot 2sqrt(3)-8/3 choice b
I got the same thing. Working out the other answers, they were either negative or obviously too large of an area for the given bounds, thank you Nade!
To find the area of the region bounded by the given curves, we first need to find the points of intersection. Let's set up the equations and solve for x.
1. The first two curves are y = x^(-1/2) and y = x^(-2). Setting them equal to each other:
x^(-1/2) = x^(-2)
2. Multiply both sides by x^2 to eliminate the negative exponents:
x^(3/2) = 1
3. Take the reciprocal of both sides to solve for x:
1/x^(3/2) = 1
x^(3/2) = 1
4. Take the square of both sides to solve for x:
x^3 = 1
x = 1
Now we have the first intersection point, which is (1,1).
Next, let's find the other intersection point.
1. The third curve is y = 1. Setting it equal to y = x^(-1/2):
1 = x^(-1/2)
2. Take the reciprocal of both sides to solve for x:
1/x^(1/2) = 1
x^(1/2) = 1
3. Square both sides to solve for x:
x = 1
Now we have the second intersection point, which is also (1,1).
Finally, let's find the last intersection point.
1. The fourth curve is y = 3. Setting it equal to y = x^(-2):
3 = x^(-2)
2. Taking the reciprocal of both sides:
1/3 = x^2
3. Taking the square root of both sides:
x = sqrt(1/3)
Now we have the third intersection point, which is approximately (0.58, 3).
To find the area, we need to integrate the difference between the curves from x = sqrt(1/3) to x = 1.
The integral for the area is given by:
A = ∫[sqrt(1/3), 1] (y_upper - y_lower) dx
Where y_upper is the upper curve (x^(-1/2) or 1) and y_lower is the lower curve (x^(-2) or 3).
Calculating the integral, we get:
A = ∫[sqrt(1/3), 1] (1 - 3) dx
A = -2x | [sqrt(1/3), 1]
A = -2(1) - (-2(sqrt(1/3)))
A = -2 + 2(sqrt(1/3))
A = 2(sqrt(1/3)) - 2
After simplifying, we find that the area of the region bounded by the given curves is 2(sqrt(1/3)) - 2.
Comparing this result to the given options, we can see that the correct answer is:
a.) 1/2(sqrt(3)) + 4/3