Find the definite integral that represents the arc length of the curve y=sqrt(x) over the interval [0, 3]
To find the definite integral that represents the arc length of the curve y = sqrt(x) over the interval [0, 3], you can use the formula for arc length:
L = ∫[a,b] sqrt(1 + (dy/dx)^2) dx
First, we need to calculate the derivative dy/dx of the given function y = sqrt(x). To do this, we can apply the power rule of differentiation.
dy/dx = (1/2) * x^(-1/2)
Next, we substitute the derivative into the arc length formula:
L = ∫[0,3] sqrt(1 + ((1/2) * x^(-1/2))^2) dx
Simplifying the expression inside the square root:
L = ∫[0,3] sqrt(1 + (1/4) * x^(-1)) dx
Now we need to integrate the expression. Let's rewrite the integral as:
L = ∫[0,3] sqrt(4/x + 1) dx
To integrate this, we will use the substitution method. Let u = (4/x + 1), and find the derivative du/dx.
du/dx = -4/x^2
Solving for dx, we get:
dx = -(x^2/4) du
Now, substitute the values of x and dx into the integral:
L = ∫ sqrt(u) * (-(x^2/4)) du
To determine the new limits of integration, we substitute x = 0 and x = 3 into the expression for u:
When x = 0, u = (4/0 + 1) = ∞
When x = 3, u = (4/3 + 1) = (7/3)
Therefore, the new limits of integration are [∞, 7/3].
Applying these limits to the integral:
L = ∫[∞,7/3] sqrt(u) * (-(x^2/4)) du
Simplifying further:
L = -(1/4) * ∫[∞,7/3] sqrt(u) * x^2 du
Now we need to evaluate this integral from ∞ to 7/3. However, integrating from ∞ is not possible, so the integral is undefined. This is because the curve y = sqrt(x) is not defined for x-values less than or equal to 0.
Therefore, the definite integral representing the arc length of the curve y = sqrt(x) over the interval [0, 3] is undefined.
S=definite integral(a÷b)sqroot[1+(dy/dx)^2]
dy/dx=1/2sqroot(x)
(dy/dx)^2=1/4x
S=definite integral(0÷3)sqroot(1+1/4x)
If you don't know how to solve that integral in google type:
wolfram alpha
When you see list of results click on:
Wolfram Alpha:Computation Knowweledge Engine
When page be open in rectangle type:
sqroot(1+1/4x)
and click =
After few seconds you will see everything about that expresion.
In rectangle Indefinite integral:
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