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Q20E

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Found in: Page 771

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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# Evaluate the line integral$$\int_{\rm{C}} {\rm{F}} {\rm{ \times dr}}$$ where $${\rm{C}}$$is given by the vector function$${\rm{r(t)}}$$.$$\begin{array}{c}{\rm{F(x,y,z) = (x + y)i + (y - z)j + }}{{\rm{z}}^{\rm{2}}}{\rm{k,}}\\{\rm{r(t) = }}{{\rm{t}}^{\rm{2}}}{\rm{i + }}{{\rm{t}}^{\rm{3}}}{\rm{j + }}{{\rm{t}}^{\rm{2}}}{\rm{k,}}\;\;\;{\rm{0}} \le {\rm{t}} \le {\rm{1}}\end{array}$$

The value of the line integral is $$\frac{{{\rm{17}}}}{{{\rm{15}}}}$$.

See the step by step solution

## Step 1: Finding  Derivative.

$${\rm{r'(t) = 2ti + 3}}{{\rm{t}}^{\rm{2}}}{\rm{j + 2tk}}$$

$$\begin{array}{l}\int_{\rm{C}} {\rm{F}} \cdot {\rm{ds = }}\int_{\rm{C}} {\rm{F}} \left( {{\rm{r(t)}} \cdot {\rm{r'(t)dt}}} \right.\\\end{array}$$

## Step 2: Integrating the Vector field.

\begin{aligned}\int_{\rm{0}}^{\rm{1}} {\left( {{{\rm{t}}^{\rm{2}}}{\rm{ + }}{{\rm{t}}^{\rm{3}}}{\rm{,}}{{\rm{t}}^{\rm{3}}}{\rm{ - }}{{\rm{t}}^{\rm{2}}}{\rm{,}}{{\rm{t}}^{\rm{4}}}} \right)} \cdot \left( {{\rm{2t,3}}{{\rm{t}}^{\rm{2}}}{\rm{,2t}}} \right)&{\rm{ = }}\int_{\rm{0}}^{\rm{1}} {\rm{2}} {{\rm{t}}^{\rm{3}}}{\rm{ + 2}}{{\rm{t}}^{\rm{4}}}{\rm{ + 3}}{{\rm{t}}^{\rm{5}}}{\rm{ - 3}}{{\rm{t}}^{\rm{4}}}{\rm{ + 2}}{{\rm{t}}^{\rm{5}}}{\rm{dt}}\\&{ = }\int_{\rm{0}}^{\rm{1}} {\rm{5}} {{\rm{t}}^{\rm{5}}}{\rm{ - }}{{\rm{t}}^{\rm{4}}}{\rm{ + 2}}{{\rm{t}}^{\rm{3}}}{\rm{dt}}\\&{ = }\frac{{{\rm{17}}}}{{{\rm{15}}}}\end{aligned}

Thus, the value of the line integral is $$\frac{{{\rm{17}}}}{{{\rm{15}}}}$$.

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