How can I find the exact length of the polar curve defined by the equation r = 82 + 0.8θ?

by

To find the exact length of the polar curve defined by the equation r = 82 + 0.8θ, we can use the formula for the length of a polar curve.

The formula for the length L of a polar curve from an angle θ = a to θ = b is given by:

L = ∫ab √((dr/dθ)2 + r2) dθ

In this case, we first need to compute dr/dθ for the given function:

r = 82 + 0.8θ

dr/dθ = 0.8

Now we can plug this and our expression for r into the length formula. Before doing that, we also need to define the limits of integration. If we do not have specific angles given, we can consider a complete interval depending on the context of the problem.

Let’s say we want to find the length from θ = 0 to θ = 10:

Now substituting everything in:

L = ∫010 √((0.8)2 + (82 + 0.8θ)2) dθ

Calculating (0.8)2 gives us 0.64. Therefore:

L = ∫010 √(0.64 + (82 + 0.8θ)2) dθ

Next, we can expand (82 + 0.8θ)2:

(82 + 0.8θ)2 = 822 + 2(82)(0.8θ) + (0.8θ)2 = 6724 + 131.2θ + 0.64θ2

So our integral becomes:

L = ∫010 √(0.64 + 6724 + 131.2θ + 0.64θ2) dθ = ∫010 √(6724.64 + 131.2θ + 0.64θ2) dθ

This integral might be complex to solve analytically, so you can consider using numerical methods such as Simpson’s rule or integration calculators to arrive at the final length value.

To summarize, the length of the polar curve can be evaluated using the above steps, and depending on the limits chosen for θ, you can find the specific length of the curve in that range.

Leave a Comment