Find the arc length of the graph of the function over the indicated interval. (Round your answer to three decimal places.)
y = (2/3)x^(3/2) +7 ;[0,6]
To find the arc length of the graph of a function over an interval, we can use the arc length formula:
L = ∫(a to b) √(1 + (dy/dx)^2) dx
Let's find the derivative of the function:
dy/dx = d/dx [(2/3)x^(3/2) + 7]
= (2/3)(3/2)x^(3/2 - 1)
= x^(1/2)
Now, let's substitute the derivative into the arc length formula and solve the integral:
L = ∫(0 to 6) √(1 + (x^(1/2))^2) dx
= ∫(0 to 6) √(1 + x) dx
To solve this integral, let's use a substitution. Let u = 1 + x, then du = dx.
When x = 0, u = 1 + 0 = 1.
When x = 6, u = 1 + 6 = 7.
Now, let's substitute u and du back into the integral:
L = ∫(1 to 7) √u du
= ∫(1 to 7) u^(1/2) du
= (2/3) * u^(3/2) |(1 to 7)
= (2/3) * (7^(3/2) - 1^(3/2))
≈ 8.533
Therefore, the arc length of the graph of the function y = (2/3)x^(3/2) + 7 over the interval [0, 6] is approximately 8.533 (rounded to three decimal places).
To find the arc length of a graph over an interval, you can use the formula:
L = ∫ √(1 + (dy/dx)^2) dx
Here's how you can apply this formula to find the arc length of the graph of the given function y = (2/3)x^(3/2) + 7 over the interval [0, 6]:
1. First, find the derivative of the function y with respect to x (dy/dx).
dy/dx = d/dx ((2/3)x^(3/2) + 7)
= (2/3)(3/2)x^(3/2 - 1)
= (1/3)x^(1/2)
2. Square the derivative: (dy/dx)^2 = [(1/3)x^(1/2)]^2 = (1/9)x.
3. Calculate the integral of √(1 + (dy/dx)^2) over the given interval [0, 6]:
L = ∫[0,6] √(1 + (1/9)x) dx
Unfortunately, finding the integral of this function is not straightforward and would involve more advanced techniques, such as integration by substitution or integration by parts. It cannot be calculated using basic integration techniques or elementary functions.
Thus, the arc length of the graph of the function y = (2/3)x^(3/2) + 7 over the interval [0, 6] cannot be determined using elementary methods alone.