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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Activity 13.3.4 .
Let
\(\vF(x,y) = \langle y^2,2xy+3\rangle\text{.}\)
(a) Let
\(C_1\) be the portion of the graph of
\(y=2x^3+3x^2-12x-15\) from
\((-2,5)\) to
\((3,30)\text{.}\) Calculate
\(\int_{C_1}\vF\cdot d\vr\text{.}\)
(b) Let
\(C_2\) be the line segment from
\((-2,5)\) to
\((3,30)\text{.}\) Calculate
\(\int_{C_2}\vF\cdot d\vr\text{.}\)
(c) Let
\(C_3\) be the circle of radius
\(3\) centered at the origin, oriented counterclockwise. Calculate
\(\oint_{C_3} \vF\cdot d\vr\text{.}\)
(d) To connect the previous parts of this activity, use a graphing utility to plot the curves
\(C_1\) and
\(C_2\) on the same axes.
(i) What type of curve is
\(C_1 - C_2\text{?}\)
(ii) What is the value of
\(\oint_{C_1-C_2}\vF\cdot d\vr\text{?}\)
(iii) What does your answer to
part c allow you to say about the value of the line integral of
\(\vF\) along the top half of
\(C_3\) compared to the line integral of
\(\vF\) from
\((3,0)\) to
\((-3,0)\) along the bottom half of the circle of radius
\(3\) centered at the origin?
