x=7t−t^2,y=4t^(3/2)
from the point (0,0) to the point (12,32), you'd have to compute
�integral b to a f(t)dt
where
a=
b=
f(t) =
To compute the integral of f(t) from point (0,0) to point (12,32), we need to find the limits of integration a and b, as well as the function f(t).
Given the parametric equations:
x = 7t - t^2
y = 4t^(3/2)
To find the limits of integration, we need to determine the values of t at both points (0,0) and (12,32).
At point (0,0), we substitute x = 0 and y = 0 into the equations:
0 = 7t - t^2
0 = 4t^(3/2)
Simplifying the second equation:
0 = 4√t^3
0 = √t^3
0 = t^(3/2)
Since t cannot be negative or complex in this case, we can conclude that t = 0 at point (0,0).
At point (12,32), we substitute x = 12 and y = 32:
12 = 7t - t^2
32 = 4t^(3/2)
To solve these equations, we can rearrange the first equation:
t^2 - 7t + 12 = 0
Factoring the quadratic equation:
(t - 3)(t - 4) = 0
So, t = 3 or t = 4.
We have found the values of t at both points, t = 0 and t = 3 or t = 4.
Now, let's determine the function f(t) for integration.
From the given equations:
f(t) = y = 4t^(3/2)
Now we have all the necessary information:
a = 0 (from the point (0,0))
b = 3 or b = 4 (from the point (12,32))
f(t) = 4t^(3/2)
To compute the integral of f(t) from b to a, we get:
∫(b to a) f(t) dt = ∫(3 or 4 to 0) 4t^(3/2) dt
Now you can evaluate this integral using the appropriate integration technique or a calculator to find the final result.