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Essential Calculus: Early Transcendentals
Found in: Page 771
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

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

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Short Answer

Use a calculator or CAS to evaluate the line integral correct to four decimal places.

\(\int_{\rm{c}} {{\rm{z}}{{\rm{e}}^{{\rm{ - xy}}}}} {\rm{ds}}\), where \({\rm{C}}\) has parametric equations,

\(\begin{array}{c}{\rm{x = t}}\\{\rm{y = }}{{\rm{t}}^{\rm{2}}}\\{\rm{z = }}{{\rm{e}}^{{\rm{ - t}}}}\end{array}\)

\({\rm{0}} \le {\rm{t}} \le 1\)

Approximate value of integral is \( \approx {\rm{0}}{\rm{.8208}}\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Find the element of the arc length.

The respective derivatives of the parametric equations are:

\(\begin{array}{c}{{\rm{x}}^\prime }{\rm{(t) = 1,}}\\{{\rm{y}}^\prime }{\rm{(t) = 2t}}\\{{\rm{z}}^\prime }{\rm{(t) = - }}{{\rm{e}}^{{\rm{ - t}}}}\end{array}\)

Element of arc length,

\(\begin{array}{c}{\rm{ds = }}\sqrt {{{\left( {{{\rm{x}}^\prime }{\rm{(t)}}} \right)}^{\rm{2}}}{\rm{ + }}{{\left( {{{\rm{y}}^\prime }{\rm{(t)}}} \right)}^{\rm{2}}}{\rm{ + }}{{\left( {{{\rm{z}}^\prime }{\rm{(t)}}} \right)}^{\rm{2}}}} \\{\rm{ = }}\sqrt {{\rm{1 + 4}}{{\rm{t}}^{\rm{2}}}{\rm{ + }}{{\rm{e}}^{{\rm{ - 2t}}}}} \end{array}\)

Step 2: Determine the value of integral.

\(\begin{array}{c}{\rm{e}}\int_{\rm{0}}^{\rm{1}} {{e^{ - t - {t^3}}}\sqrt {{{\left( {{x^\prime }\left( t \right)} \right)}^2} + {{\left( {{y^\prime }\left( t \right)} \right)}^2} + {{\left( {{z^\prime }\left( t \right)} \right)}^2}} } \\{\rm{ = }}\sqrt {{\rm{1 + 4}}{{\rm{t}}^{\rm{2}}}{\rm{ + }}{{\rm{e}}^{{\rm{ - 2t}}}}} \\ \approx {\rm{0}}{\rm{.8208}}\end{array}\)

Therefore, approximate value of integral is \( \approx {\rm{0}}{\rm{.8208}}\).

Most popular questions for Math Textbooks

(a). To show: The parametric equations \(x = a\cosh u\cos v,y = b\cosh u\sin v,z = c\sinh u\) represent a hyperboloid of one sheet.

(b). To draw: The graph of a hyperboloid of one sheet with the parametric equations \(x = a\cosh u\cos v,y = b\cosh u\sin v,z = c\sinh u\) for \(a = 1,b = 2\), and \(c = 3\).

(c). To find: The expression for surface area of the hyperboloid of one sheet with the parametric equations \(x = a\cosh u\cos v,y = b\cosh u\sin v,z = c\sinh u\) for \(a = 1,b = 2\), and \(c = 3\).

\({\bf{F}}(x,y,z) = {x^4}{\bf{i}} - {x^3}{z^2}{\bf{j}} + 4x{y^2}z{\bf{k}}\), \(S\)is the surface of the solid bounded by the cylinder \({x^2} + {y^2} = 1\) and the planes \(z = x + 2\) and \(z = 0\).

To determine the value of for \({\rm{F}}(x,y,z) = xzi + xj + y{\rm{k}}\) is \(0\).

If you have a CAS that plots vector fields (the command is field plot in Maple and Plot Vector Field or Vector Plot in Mathematica), use it to plot\({\bf{F}}(x,y) = \left( {{y^2} - 2xy} \right){\bf{i}} + \left( {3xy - 6{x^2}} \right){\bf{j}}\)

Explain the appearance by finding the set of points \((x,y)\) such that \({\bf{F}}(x,y) = {\bf{0}}\).

Evaluate the line integral, where C is the given curve.

\(\) \(\int_{\rm{C}} {\rm{x}} {\rm{ sin y ds}}\), C is the line segment from \({\rm{(0,}}\mid {\rm{3) to (4,6)}}\).
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