Final answer:
The length of the polar curve r=θ² from θ=0 to θ=√5 is found by computing the integral of the square root of (r² + (dr/dθ)²) over the given range.
Step-by-step explanation:
The student is asking for the length of the polar curve given by the equation r=θ², where θ (theta) ranges from 0 to √5. To find the length of a polar curve, given by r=f(θ), from θ=a to θ=b, we use the integral ∫_a^b √(r² + (dr/dθ)²) dθ.
In this case, as r=θ², the derivative dr/dθ is 2θ. Plugging these into the formula, we get the integral ∫_0^√5 √((θ²)² + (2θ)²) dθ, which simplifies to ∫_0^√5 √(θ^4 + 4θ²) dθ. This integral will give us the arc length of the polar curve from θ=0 to θ=√5.