Vault Of Euclid

Mass of a Helical Spring

Multivariable calculus core

Calculate the mass of a spring in the shape of a helix parameterized by \( \mathbf{r}(t) = \langle \cos t, \sin t, t \rangle, 0 \le t \le 6\pi \), with a density function given by \( \rho(x, y, z) = x + y + z \) kg/m.
calculus-iiiline-integralscalar-line-integralmassdensityhelixarc-lengthvector-functionstrigonometric-functionsdefinite-integralevaluate-integralfind-mass

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What this problem tests

Tests the ability to set up and evaluate a scalar line integral over a 3D curve, including finding the arc length differential and integrating trigonometric functions.

This is a standard application of scalar line integrals in a multivariable calculus course.

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