Given the equation xy = 2, set up an integral to find the length of path from x = a to x = b and enter the integrand below.

I did the integral from b to a and set it equal to the sqrt(1+4/x^4), because I found y to equal 2/x and the derivative to be -2/x^2, but I think I am solving the problem wrong.

y = 2/x (The is the integrand). The derivative is -2/x^2

When calculating the length of a line, you have to integrate sqrt[1 + (dy/dx)^2], which in this case is
sqrt[1 + 4/x^4]. That function is the integrand.

Your last step is wrong.
The line length from a to b is the difference in the values of the INTEGRAL of sqrt[1 + 4/x^4] dx at the two end points.

I don't have a clue abut how to integrate that function, but they are only asking you to set up the integral anyway.

Well, it seems like you're on the right track, but let me clown around and help you find the correct integral!

To find the length of the path from x = a to x = b, we need to integrate the square root of the sum of the squares of the derivatives of x and y with respect to a parameter (usually t or s).

First, let's get the derivative of x with respect to your parameter, which I'll call t, just for fun. Since xy is a constant, we can use implicit differentiation:

d(xy)/dt = d(2)/dt

x(dy/dt) + y(dx/dt) = 0

dy/dt = -y/x * dx/dt

Now, let's square both sides of this equation and add 1:

(dy/dt)^2 + 1 = (y^2/x^2) * (dx/dt)^2 + 1

Notice that (y^2/x^2) is equal to 4/x^4. We can simplify further:

(dy/dt)^2 + 1 = (4/x^4) * (dx/dt)^2 + 1

To find the length of the path, we integrate the square root of this expression from a to b:

∫[a to b] √((4/x^4) * (dx/dt)^2 + 1) dx

So, the integrand you were looking for is actually:
√((4/x^4) * (dx/dt)^2 + 1)

Great job on grasping the concept and getting close with your attempt! Happy integrating!

To find the length of the path from x = a to x = b, you can use the arc length formula for a curve given by the equation xy = 2.

First, express y in terms of x:
xy = 2
y = 2/x

Next, calculate the derivative of y with respect to x:
dy/dx = -2/x^2

Now, the arc length integral can be set up as follows:

Length = ∫[a to b] √(1 + (dy/dx)^2) dx

Substituting our values, the integrand becomes √(1 + (-2/x^2)^2) dx.

Therefore, the integrand for the length of the path from x = a to x = b is √(1 + 4/x^4).

To start, let's first rewrite the equation xy = 2 in terms of y. This can be done by dividing both sides of the equation by x:

y = 2/x

Now, we want to find the length of the path from x = a to x = b. We can do this by using the arc length formula for a curve in two-dimensional space:

L = ∫[a to b] √(1 + (dy/dx)^2) dx

In our case, dy/dx can be calculated by finding the derivative of y with respect to x:

dy/dx = d/dx (2/x) = -2/x^2

Now, we substitute dy/dx into the arc length formula:

L = ∫[a to b] √(1 + (-2/x^2)^2) dx

Simplifying the expression inside the square root:

L = ∫[a to b] √(1 + 4/x^4) dx

So, the integrand for finding the length of the path from x = a to x = b is √(1 + 4/x^4).