Question

(1 point) Suppose \(\vec{F}(x, y) = x^2 \vec{i} + y^2 \vec{j}\) and C is the line segment segment from point P = (0, -4) to Q = (-3, 1). (a) Find a vector parametric equation \(\vec{r}(t)\) for the line segment C so that points P and Q correspond to t = 0 and t = 1, respectively. \(\vec{r}(t) = (0, -4) + t(-3, -3)\) (b) Using the parametrization in part (a), the line integral of \(\vec{F}\) along C is \(\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) dt = \int_a^b \text{} dt\) with limits of integration a = and b = (c) Evaluate the line integral in part (b).

          (1 point) Suppose \(\vec{F}(x, y) = x^2 \vec{i} + y^2 \vec{j}\) and C is the line segment segment from point P = (0, -4) to Q = (-3, 1).
(a) Find a vector parametric equation \(\vec{r}(t)\) for the line segment C so that points P and Q correspond to t = 0 and t = 1, respectively.
\(\vec{r}(t) = (0, -4) + t(-3, -3)\)
(b) Using the parametrization in part (a), the line integral of \(\vec{F}\) along C is
\(\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) dt = \int_a^b \text{________} dt\)
with limits of integration a = ______ and b = ______
(c) Evaluate the line integral in part (b).

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(1 point) Suppose F⃗(x, y) = x^2 i⃗ + y^2 j⃗ and C is the line segment segment from point P = (0, -4) to Q = (-3, 1).
(a) Find a vector parametric equation r⃗(t) for the line segment C so that points P and Q correspond to t = 0 and t = 1, respectively.
r⃗(t) = (0, -4) + t(-3, -3)
(b) Using the parametrization in part (a), the line integral of F⃗ along C is
F⃗· dr⃗ = ^b F⃗(r⃗(t)) ·r⃗'(t) dt = ^b dt
with limits of integration a = and b =
(c) Evaluate the line integral in part (b).

Added by Rhonda H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Supreeta N

08/11/2023

Step 1

Step 1: To find the vector parametric equation r(t) for the line segment C, we can use the formula r(t) = P + t(Q - P), where P and Q are the given points and t varies from 0 to 1. Show more…

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(1 point) Suppose vec(F)(x,y)=x^(2)vec(i)+y^(2)vec(j) and C is the line segment segment from point P=(0,-4) to Q=(-3,1). (a) Find a vector parametric equation vec(r)(t) for the line segment C so that points P and Q correspond to t=0 and t=1, respectively. vec(r)(t)= (b) Using the parametrization in part (a), the line integral of vec(F) along C is int_C vec(F)*dvec(r)=int_a^b vec(F)(vec(r)(t))*vec(r)^(')(t)dt=int_a^b dt with limits of integration a= and b= (c) Evaluate the line integral in part (b). (1 point) Suppose F(, y) = 2i + y2j and C is the line segment segment from point P = (0, -4) to Q =(-3,1). (a) Find a vector parametric equation r(t) for the line segment C so that points P and Q correspond to t=0 and t=1,respectively r(t) (0,-4) +t(-3,-3) (b) Using the parametrization in part (a), the line integral of F along C is F(r(t)).r'(t)dt= dt with limits of integration a : and b (c) Evaluate the line integral in part (b)

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