Chapter 14: Q 12. (page 1154)

Evaluate the line integral of the given function over the specified curve.
$\int_{c} F (x, y) . d r$ where $F (x, y) = (8 x^{2} y, - 9 x y^{2})$ and C is the curve parametrized by $x = t^{2}, y = t^{3}$ for $0 \leq t \leq 1$ .

Short Answer

Expert verified

$\int_{c} F (x, y) . d r = - \frac{67}{99}$

Step by step solution

Given Information

The given expression is $F (x, y) = (8 x^{2} y, - 9 x y^{2})$ and $x = t^{2}, y = t^{3}$ and $0 \leq t \leq 1$

Finding the line integral

The curve is parametrized as

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

Differentiating with respect to t,

$r^{'} (t) = (2 t, 3 t^{2})$

Using vales of $x, y$ in $F (x, y)$ , we get

$F (x (t), y (t)) = (8 {(t^{2})}^{2} . t^{3}, - 9 t^{2} {(t^{2})}^{2})$

$F (x (t), y (t)) = (8 t^{7} . - 9 t^{8})$

The line integral is evaluated as

localid="1653379504558" $\int_{c} F (x, y) \cdot d r = \int_{c} F (x (t), y (t)) \cdot r^{'} (t) d t$

localid="1653379508896" $\int_{c} F (x, y) \cdot d r = \int_{0}^{1} (8 t^{7}, - 9 t^{8}) \cdot (2 t, 3 t^{2}) d t$

Solving dot product

localid="1653379515036" $= \int_{0}^{1} (8 t^{7} (2 t) + (- 9 t^{8}) \cdot 3 t^{2}) d t$

localid="1652329204037" $= \int_{0}^{1} (16 t^{8} - 27 t^{10}) d t$

localid="1652329261962" $= 16 {|\frac{t^{9}}{9}|}^{1}_{0} - 27 {|\frac{t^{11}}{11}|}^{1}_{0}$

localid="1652329246329" $= [\frac{16}{9} - \frac{27}{11}] - 0$

localid="1652329278609" $= - \frac{67}{99}$

Hence, $\int_{c} F (x, y) \cdot d r = - \frac{67}{99}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Evaluate the line integral of the given function over the specified curve.
$\int_{c} F (x, y) . d r$ where $F (x, y) = (8 x^{2} y, - 9 x y^{2})$ and C is the curve parametrized by $x = t^{2}, y = t^{3}$ for $0 \leq t \leq 1$ .

Short Answer

Step by step solution

Given Information

Finding the line integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

Evaluate the line integral of the given function over the specified curve.∫cFx,y.drwhere Fx,y=8x2y,-9xy2and C is the curve parametrized by x=t2,y=t3for 0≤t≤1.

Short Answer

Step by step solution

Given Information

Finding the line integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Geometry

Logic and Functions

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 $F (x, y) = (8 x^{2} y, - 9 x y^{2})$ and C is the curve parametrized by $x = t^{2}, y = t^{3}$ for $0 \leq t \leq 1$ .