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Chapter 14: Q 12. (page 1154)

Evaluate the line integral of the given function over the specified curve.
${鈭�}_{c} F (x, y) . d r$ 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 $0 鈮� t 鈮� 1$ .

Short Answer

Expert verified

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

Step by step solution

01

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 $0 鈮� t 鈮� 1$

02

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" ${鈭�}_{c} F (x, y) 路 d r = {鈭�}_{c} F (x (t), y (t)) 路 r^{'} (t) d t$

localid="1653379508896" ${鈭�}_{c} F (x, y) 路 d r = {鈭�}_{0}^{1} (8 t^{7}, - 9 t^{8}) 路 (2 t, 3 t^{2}) d t$

Solving dot product

localid="1653379515036" $= {鈭�}_{0}^{1} (8 t^{7} (2 t) + (- 9 t^{8}) 路 3 t^{2}) d t$

localid="1652329204037" $= {鈭�}_{0}^{1} (16 t^{8} - 27 t^{10}) d t$

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, ${鈭�}_{c} F (x, y) 路 d r = - \frac{67}{99}$

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