Chapter 14: Q 16. (page 1154)

Evaluate the line integral of the given function over the specified curve.
$\int_{C} F (x, y, z) . d r$ where $F (x, y, z) = (e^{x}, e^{y}, e^{z})$ and C is the curve parametrized by $x = t, y = t^{2}, z = t^{3}$ for $0 \leq t \leq 1$ .

Short Answer

Expert verified

$\int_{C} F (x, y, z) . d r = 3 e$

Step by step solution

Given Information

The given expression is $F (x, y, z) = (e^{x}, e^{y}, e^{z})$ and $x = t, y = t^{2}, z = t^{3}$

Simplification

Using the given values of $x, y, z$ in function

$F (x, y, z) = (e^{x}, e^{y}, e^{z})$

Differentiating parameters, we get

$r (t) = (1, 2 t, 3 t^{2})$

The line integral is evaluated as

\int_{C} F (x, y, z) . d r = \int_{0}^{1} (e^{x} e^{y} e^{z}) (1, 2 t, 3 t^{2}) d t

$\int_{C} F (x, y, z) . d r = \int_{0}^{1} (e^{t} + 2 t e^{t^{2}} + 3 t^{2} e^{t^{3}}) d t$

localid="1651521623828" $\int_{C} F (x, y, z) . d r = {|e^{t} + e^{t^{2}} + e^{t^{3}}|}_{0}^{1}$

(This is obtained by solving integration by substitution)

$\int_{C} F (x, y, z) . d r = e + e + e = 3 e$

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Evaluate the line integral of the given function over the specified curve.
$\int_{C} F (x, y, z) . d r$ where $F (x, y, z) = (e^{x}, e^{y}, e^{z})$ and C is the curve parametrized by $x = t, y = t^{2}, z = t^{3}$ for $0 \leq t \leq 1$ .

Short Answer

Step by step solution

Given Information

Simplification

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Evaluate the line integral of the given function over the specified curve.∫CFx,y,z.drwhere Fx,y,z=ex,ey,ezand C is the curve parametrized by x=t,y=t2,z=t3for 0≤t≤1.

Short Answer

Step by step solution

Given Information

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Logic and Functions

Pure Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Evaluate the line integral of the given function over the specified curve.
$\int_{C} F (x, y, z) . d r$ where $F (x, y, z) = (e^{x}, e^{y}, e^{z})$ and C is the curve parametrized by $x = t, y = t^{2}, z = t^{3}$ for $0 \leq t \leq 1$ .