Find the exact length of the curve.
x = 1 + 9^t2, y = 7 + 6^t3, 0 ≤ t ≤ 2
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Find the exact length of the curve.
x = 1 + 9t^2, y = 7 + 6t^3, 0 ≤ t ≤ 2
s = ∫[0,2] √((18t)^2 + (18t^2)^2) dt
= ∫[0,2] 18t√(1+t^2) dt = 6(5√5 - 1)
To find the exact length of a curve, we can use the concept of arc length. The arc length formula for a parametric curve is given by:
L = ∫[a, b] √(dx/dt)^2 + (dy/dt)^2 dt
In this case, we are given the parametric equations x = 1 + 9^t^2 and y = 7 + 6^t^3, where 0 ≤ t ≤ 2. To find the exact length of the curve, we need to find the derivatives dx/dt and dy/dt, and then evaluate the integral.
Step 1: Find the derivatives dx/dt and dy/dt.
To find dx/dt, differentiate x = 1 + 9^t^2 with respect to t:
dx/dt = 2t * 9^t^2-1
To find dy/dt, differentiate y = 7 + 6^t^3 with respect to t:
dy/dt = 3t^2 * 6^t^3-1
Step 2: Evaluate the integral.
Now that we have found dx/dt and dy/dt, we can substitute them into the arc length formula:
L = ∫[0, 2] √(2t * 9^t^2-1)^2 + (3t^2 * 6^t^3-1)^2 dt
Unfortunately, the integral given is quite complex and does not have a closed-form solution. Therefore, to find the exact length of the curve, we would need to use numerical methods or a computer program to approximate the value of the integral.