Mathematics

Mathematics

Problem

Calculate the exact length of the curve y=x22y=\dfrac{x^{2}}{2} on the interval [−1,1][-1,1]. a) 222\sqrt{2} b) 2+2ln⁡(1+2)\sqrt{2}+2\ln\bigl(1+\sqrt{2}\bigr) c) 2+ln⁡(1+2)\sqrt{2}+\ln\bigl(1+\sqrt{2}\bigr) d) ln⁡(1+2)\ln\bigl(1+\sqrt{2}\bigr)
Intuitive explanation

Arc length asks: if you traced along a curve, how far would you actually travel? The key is the arc length formula L=∫ab1+(y′)2 dxL = \int_a^b \sqrt{1 + (y')^2} \, dx, which accounts for both horizontal and vertical movement. Here, you'll find y′y', substitute into the formula, and simplify the expression under the square root—look for a perfect square or a recognizable form that makes the integral tractable.

Intuitive explanation

Arc length asks: if you traced along a curve, how far would you actually travel? The key is the arc length formula L=∫ab1+(y′)2 dxL = \int_a^b \sqrt{1 + (y')^2} \, dx, which accounts for both horizontal and vertical movement. Here, you'll find y′y', substitute into the formula, and simplify the expression under the square root—look for a perfect square or a recognizable form that makes the integral tractable.

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