Let ƒ(x, y) = x² + 4y² and let C be the line segment from (0, 0) to (2, 2). You are going to compute Jo Vf. dr two ways: first, using the method learned in section 6.2 for evaluating line integrals, and second, using the fundamental theorem for line integrals. First way: Vf=( C can be parameterized by r(t) Then 7'(t): and ▼ ƒ(r(t)) = ( So = || 10 = 2 = 1² = ( Vf.dr [²* ▼ ƒ (F(t)) - 7"' (t) dt dt = (t, " ). > ) for 0 ≤ t ≤ 2. ).
Let ƒ(x, y) = x² + 4y² and let C be the line segment from (0, 0) to (2, 2). You are going to compute Jo Vf. dr two ways: first, using the method learned in section 6.2 for evaluating line integrals, and second, using the fundamental theorem for line integrals. First way: Vf=( C can be parameterized by r(t) Then 7'(t): and ▼ ƒ(r(t)) = ( So = || 10 = 2 = 1² = ( Vf.dr [²* ▼ ƒ (F(t)) - 7"' (t) dt dt = (t, " ). > ) for 0 ≤ t ≤ 2. ).
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Transcribed Image Text:Second way (using Fundamental Theorem for Line Integrals):
ř (0) = (
),
so f(7(0)) =
and 7 (2) = (
so f(r (2))
Then
2
là ▼ƒ·dr = f(r(2)) — ƒ(ƒ(0))
=
),
Transcribed Image Text:Let f(x, y) = x² + 4y² and let C be the line segment from (0, 0) to (2, 2).
You are going to compute
là
Vf. dr two ways: first, using the method learned in section 6.2 for
с
evaluating line integrals, and second, using the fundamental theorem for line integrals.
First way:
Vf=(
C can be parameterized by r(t) = (t,
Then '(t)
and ▼ ƒ(r(t)) = {
so sv.
So
2
= [²
=
2
- 1²
||
(
Vf. dr
▼ f(r(t)). r' (t)dt
dt
).
>
> for 0 ≤ t ≤ 2.
).
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