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Consider the curve C: r(t) = cost i + sin t j + e^t k, \space 0 \leq t \leq 1. (a) Express the...

Question:

Consider the curve C: {eq}r(t) = cost i + sin t j + e^t k, \space 0 \leq t \leq 1 {/eq}.

(a) Express the length of the curve C as a definite integral. DO NOT evaluate.

(b) Evaluate the line integral with respect to arc length.

{eq}\int_C z^2 ds {/eq}

Length of a curve; line integral of a function

This example is in two parts: the first part illustrates how to express the arc length of a curve in three dimensions as a definite integral; the second part shows how to evaluate the line integral of a function with respect to arc length.

Answer and Explanation:

Part (a)

If we have the vector function {eq}\mathbf{r} = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k} {/eq}, where t is a parameter,

then the...

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Line Integrals: How to Integrate Functions Over Paths

from GRE Math: Study Guide & Test Prep

Chapter 15 / Lesson 2
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