Find the length of the curve. You may use your calculator.
f(x)=x^(1/3)+x^(2/3) [0,2]
I understand that the function needs to be written as x in terms of y because there's a vertical tangent at x=0, but I don't understand how to go about the problem other than that.
The vertical slope does not matter, once you go through the hoops. The answer is shown here:
http://www.wolframalpha.com/input/?i=plot+x%5E(1%2F3)%2Bx%5E(2%2F3)+for+x%3D0..2
The problem I have is evaluating that integral! I tried something like
u = 1+2∛x
du = 2/3 x^(-2/3)
x = (u-1)^3/8
and the integral then becomes
∫1/2 √(9(u-1)^4/8 + u^2) du
which is no easier to handle. Looks like this is really a calculator problem.
analyze how Wolfram does it.
https://www.wolframalpha.com/input/?i=arc+length+of+%3Dx%5E(1%2F3)%2Bx%5E(2%2F3)++from+x%3D0+to+x%3D2
To find the length of the curve defined by the equation f(x) = x^(1/3) + x^(2/3) over the interval [0,2], you can use the arc length formula for a curve in terms of y.
The first step is to rewrite the equation in terms of y. To do this, we need to solve the equation for x in terms of y. Let's start with the equation:
f(x) = x^(1/3) + x^(2/3)
Rearranging the equation, we get:
x^(2/3) = f(x) - x^(1/3)
Next, raise both sides of the equation to the power of 3 to eliminate the fractional exponent:
(x^(2/3))^3 = (f(x) - x^(1/3))^3
Expanding the equation, we have:
x^2 = (f(x) - x^(1/3))^3
Now, we can solve this equation for x to get x in terms of y. Since there is a vertical tangent at x = 0, we will need to consider this case separately. For x > 0, we have:
x = (f(x) - x^(1/3))^3^(1/2)
Now, we have x expressed in terms of y. We know that the domain of x is [0,2], so we need to find the corresponding range of y values for this interval. We can substitute the values of x = 0 and x = 2 into the equation to find the corresponding y values:
For x = 0, we have:
f(0) = (0 - 0^(1/3))^3^(1/2)
Simplifying, we get:
f(0) = 0
For x = 2, we have:
f(2) = (2 - 2^(1/3))^3^(1/2)
Simplifying further, we get:
f(2) = (2 - 2^(1/3))^3^(1/2)
Now that we have x expressed in terms of y and the corresponding range of y values, we can proceed with finding the length of the curve using the arc length formula. The formula is given by:
L = ∫[a,b] √(1 + (dy/dx)^2) dx
In this case, since we have x in terms of y, we need to find dy/dx. Differentiating x with respect to y, we get:
dx/dy = 1 / (dy/dx)
Now, we can use the chain rule to find dy/dx:
dy/dx = 1 / (dx/dy)
Substituting x = (f(x) - x^(1/3))^3^(1/2), we can find dx/dy:
dx/dy = 1 / (dy/dx)
Now, we can substitute dy/dx and dx/dy into the arc length formula:
L = ∫[a,b] √(1 + (dx/dy)^2) dy
Using the values we obtained earlier for f(0) and f(2), we can now solve the integral to find the length of the curve.
You can input the integral (∫) into your calculator using appropriate limits of integration [a,b] and evaluate it to get the precise length of the curve. Remember to use parentheses or brackets to indicate the limits of integration and make sure your calculator is in radian mode if necessary.