To find the length of the curve defined by y = 5x^3+9x from the point (-3,-162) to the point (1,14), you’d have to compute the integral [a,b] f(x)dx
where
a =
b =
f(x) =
a
L = ∫ [ sqrt 1 + ( y´ )^ 2 ] dx
b
In this case:
y´= 3 * 5 * x ^ 2 + 9 = 15 x ^ 2 + 9
1 + ( y´ )^ 2 = 1 + ( 15 x ^ 2 + 9 ) ^ 2
a = - 3
b = 1
f(x) = sqroot [ 1 + ( 15 x ^ 2 + 9 ) ^ 2 ]
To find the length of the curve defined by the equation y = 5x^3 + 9x from the point (-3,-162) to the point (1,14), you need to compute the integral of the function f(x) = sqrt(1 + (dy/dx)^2) with respect to x over the interval [a, b].
To calculate the integral, we need to find the derivative of y with respect to x, which gives us dy/dx.
dy/dx = d/dx(5x^3 + 9x)
= 15x^2 + 9.
Then, the function f(x) = sqrt(1 + (dy/dx)^2) becomes:
f(x) = sqrt(1 + (15x^2 + 9)^2).
Next, we need to determine the values of a and b. In this case, a = -3 and b = 1, as these are the x-coordinates of the given points.
Now, we are ready to compute the integral [a, b] f(x)dx:
∫[a,b] sqrt(1 + (15x^2 + 9)^2) dx.
You can use calculus techniques to evaluate this integral, such as substitution or integration by parts.