Chapter 14: Q 28. (page 1106)

Evaluate the multivariate line integral of the given function over the specified curve.
$f (x, y, z) = e^{x^{2} + y^{2} + z^{2}}$ , with $C$ the circular helix of radius $1$ , centered about the $y$ -axis, and parametrized by $r (t) = (\cos t, t, \sin t)$ for $π \leq y \leq 3 π$ .

Short Answer

Expert verified

The required integral is $\int_{C} (e^{x^{2} + y^{2} + z^{2}}) d s = \sqrt{2} e π^{\frac{3}{2}}$

Step by step solution

Given Information

It is given that

$f (x, y, z) = e^{x^{2} + y^{2} + z^{2}}$

Curve is parametrized by

$C : r (t) = ⟨ \cos t, t, \sin t ⟩$

Solving the parametrized curve

The parametric equations are $x (t) = \cos t, y (t) = t, z (t) = \sin t, π \leq t \leq 3 π$

$\Rightarrow x^{'} (t) = - \sin t, y^{'} (t) = 1, z^{'} (t) = \cos t$

Also

$d s = ||r^{'} (t)|| d t$

$= \sqrt{{[x^{'} (t)]}^{2} + {[y^{'} (t)]}^{2} + {[z^{'} (t)]}^{2}} d t$

$= \sqrt{(- \sin t)^{2} + (\cos t)^{2} + (1)^{2}} d t$

$= \sqrt{1 + 1} d t$

$= \sqrt{2} d t$

$\Rightarrow f (x (t), y (t), z (t)) = e^{\cos^{2} t + r^{2} + \sin^{2} t}$

$\Rightarrow f (x (t), y (t), z (t)) = e^{1 + t^{2}}$

Finding the integral

The form of line integral is

$\int_{C} f (x, y, z) d s = \int_{a}^{b} f (x (t), y (t), z (t)) ||r^{'} (t)|| d t$

$\int_{C} (e^{x^{2} + y^{2} + z^{2}}) d s = \int_{π}^{3 π} e^{1 + t^{2}} \sqrt{2} d t$

$= \sqrt{2} \cdot e \int_{π}^{3 π} e^{t^{2}} d t$

$= \sqrt{2} e \cdot π^{\frac{3}{2}}$

Hence, $\int_{C} (e^{x^{2} + y^{2} + z^{2}}) d s = \sqrt{2} e π^{\frac{3}{2}}$

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Evaluate the multivariate line integral of the given function over the specified curve.
$f (x, y, z) = e^{x^{2} + y^{2} + z^{2}}$ , with $C$ the circular helix of radius $1$ , centered about the $y$ -axis, and parametrized by $r (t) = (\cos t, t, \sin t)$ for $π \leq y \leq 3 π$ .

Short Answer

Step by step solution

Given Information

Solving the parametrized curve

Finding the integral

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Evaluate the multivariate line integral of the given function over the specified curve.f(x,y,z)=ex2+y2+z2, with Cthe circular helix of radius 1, centered about the y-axis, and parametrized byrt=(cost,t,sint)forπ≤y≤3π.

Short Answer

Step by step solution

Given Information

Solving the parametrized curve

Finding the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Pure Maths

Theoretical and Mathematical Physics

Decision Maths

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Evaluate the multivariate line integral of the given function over the specified curve.
$f (x, y, z) = e^{x^{2} + y^{2} + z^{2}}$ , with $C$ the circular helix of radius $1$ , centered about the $y$ -axis, and parametrized by $r (t) = (\cos t, t, \sin t)$ for $π \leq y \leq 3 π$ .