Chapter 14: Q 11. (page 1154)

Evaluate the line integral of the given function over the specified curve.
$\int_{C} F (x, y) . d r$ where $F (x, y) = 7 y^{2} i - 3 x y j$ and C is the line segment $x = t, y = 3 t$ for $0 \leq t \leq 1$

Short Answer

Expert verified

$\int_{C} F (x, y) . d r = 12$

Step by step solution

Given Information

The given expression is $F (x, y) = 7 y^{2} i - 3 x y j, x = t, y = 3 t$ localid="1652874571194" $x = t, y = 3 t$ .

To evaluate line integral

The curve is parametrized as,

$r (t) = (t, 3 t)$ for $0 \leq t \leq 1$

Differentiating it with respect to t,

$r^{'} (t) = (1, 3)$

Finding field vector by using values of $(x, y)$ ,

$F (x (t), y (t)) = 7 {(3 t)}^{2} i - 3 t (3 t) j$

$= 63 t^{2} i - 9 t^{2} j$

$= (63 t^{2}, - 9 t^{2})$

Line integral is calculated as,

$\int_{C} F (x, y) . d r = \int_{C} F (x (t), y (t)) . r^{'} (t) d t$

$= \int_{C} (63 t^{2}, - 9 t^{2}) \cdot (1, 3) d t$

localid="1652328520867" $= \int_{0}^{1} (63 t^{2} \cdot 1 + (- 9 t^{2}) \cdot 3) d t$ (Simplify dot product)

localid="1652956410638" $= \int_{0}^{1} (63 t^{2} - 27 t^{2}) d t$ $= \int_{0}^{1} 36 t^{2} d t$

$= {[12 t^{3}]}_{0}^{1}$

$= 12 (1) - 12 (0)$

$= 12$

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Evaluate the line integral of the given function over the specified curve.
$\int_{C} F (x, y) . d r$ where $F (x, y) = 7 y^{2} i - 3 x y j$ and C is the line segment $x = t, y = 3 t$ for $0 \leq t \leq 1$

Short Answer

Step by step solution

Given Information

To evaluate line integral

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Evaluate the line integral of the given function over the specified curve.∫CF(x,y).drwhere F(x,y)=7y2i-3xyjand C is the line segment x=t,y=3tfor0≤t≤1

Short Answer

Step by step solution

Given Information

To evaluate line integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Theoretical and Mathematical Physics

Statistics

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Evaluate the line integral of the given function over the specified curve.
$\int_{C} F (x, y) . d r$ where $F (x, y) = 7 y^{2} i - 3 x y j$ and C is the line segment $x = t, y = 3 t$ for $0 \leq t \leq 1$