Find the length of the curve y=(1/(x^2)) from ( 1, 1 ) to ( 2, 1/4 ) [set up the problem only, don't integrate/evaluate]
this is what i did.. let me know asap if i did it right..
y = (1/(x^2))
dy/dx = (-2/(x^3))
L = integral from a to b for: sqrt(1+(dy/dx)^2)dx
L = integral from 1 to 2 for: sqrt(1+(-2/(x^3))^2)dx
L = integral from 1 to 2 for: sqrt(1+(-2/(x^3))(-2/(x^3)))dx
L = integral from 1 to 2 for: sqrt(1+(4/(x^6))dx
a=1
b=2
n=1-
deltaX=0.1
f(x)=sqrt(1+(4/x^6))
L = integral from 1 to 2 for: sqrt(1+(4/(x^6))dx
L = (deltaX/3)[ f(1) + 4f(1.1) + 2f(1.2) + 4f(1.3) + ... + 2f(1.8) + 4f(1.9) + f(2) ]
L = (0.1/3)[ sqrt(1+(4/1)^6) + 4sqrt(1+(4/1.1)^6) + 2sqrt(1+(4/1.2)^6) + 4sqrt(1+(4/1.3)^6) + 2sqrt(1+(4/1.4)^6) + 4sqrt(1+(4/1.5)^6) + 2sqrt(1+(4/1.6)^6) + 4sqrt(1+(4/1.7)^6) + 2sqrt(1+(4/1.8)^6) + 4sqrt(1+(4/1.9)^6) + sqrt(1+(4/2)^6) ]
L = (0.1/3)[720.937]
L = 24.031
To find the length of the curve y = 1/(x^2) from (1, 1) to (2, 1/4), you can use the arc length formula.
1. First, find the derivative of the equation y = 1/(x^2) to get dy/dx.
dy/dx = -2/(x^3)
2. Then, use the arc length formula:
L = ∫√(1 + (dy/dx)^2) dx
where the integral is taken from the x-value of the starting point to the x-value of the ending point.
L = ∫√(1 + (-2/(x^3))^2) dx
3. Simplify the expression inside the square root:
L = ∫√(1 + (4/(x^6))) dx
4. Set up the definite integral using the limits of integration:
L = ∫[from 1 to 2] √(1 + (4/(x^6))) dx
Note: The limits of integration are determined by the x-values of the starting and ending points of the curve.
5. You can then evaluate the integral to find the length of the curve. However, please note that you have only set up the problem and have not integrated or evaluated it yet.
L = (Δx/3)[f(x1) + 4f(x2) + 2f(x3) + 4f(x4) + ... + 2f(xn-3) + 4f(xn-2) + f(xn)]
where Δx is the width of each subinterval, f(x) is the function inside the square root, and n is the number of subintervals.
In the provided example, Δx = 0.1, f(x) = √(1 + (4/(x^6))), a = 1, and b = 2. However, you haven't provided the value of n or the specific subinterval points, so the integral has not been evaluated.
Therefore, the length of the curve, as it stands, is still represented as an integral:
L = ∫[from 1 to 2] √(1 + (4/(x^6))) dx
To find the actual value of L, you would need to integrate and evaluate the expression using appropriate numerical methods or techniques.