f(x,y) = /1+9ry, y = r³ for 0 < x < 1. (line integral for a function of two variables is computed the same way as a function of three variables, except C is on the ry– plane.

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Chapter2: Second-order Linear Odes
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help on C and D please

(3) Compute \( \int_C f \, ds \) over the curve specified.

(a) \( f(x, y, z) = z^2, \, r(t) = \langle 2t, 3t, 4t \rangle \) for \( 0 \leq t \leq 2 \)

(b) \( f(x, y, z) = 2x^2 + 8z, \, r(t) = \langle e^t, t^2, t \rangle \) for \( 0 \leq t \leq 1 \)

For these last two problems find \( r(t) \) by parametrizing \( C \).

(c) \( f(x, y) = \sqrt{1 + 9xy}, \, y = x^3 \) for \( 0 \leq x \leq 1 \). (Line integral for a function of two variables is computed the same way as a function of three variables, except \( C \) is on the xy-plane.)

(d) \( f(x, y, z) = x + yz, \) and \( C \) is the line segment from \( P = (0, 0, 0) \) to \( (6, 2, 2) \).
Transcribed Image Text:(3) Compute \( \int_C f \, ds \) over the curve specified. (a) \( f(x, y, z) = z^2, \, r(t) = \langle 2t, 3t, 4t \rangle \) for \( 0 \leq t \leq 2 \) (b) \( f(x, y, z) = 2x^2 + 8z, \, r(t) = \langle e^t, t^2, t \rangle \) for \( 0 \leq t \leq 1 \) For these last two problems find \( r(t) \) by parametrizing \( C \). (c) \( f(x, y) = \sqrt{1 + 9xy}, \, y = x^3 \) for \( 0 \leq x \leq 1 \). (Line integral for a function of two variables is computed the same way as a function of three variables, except \( C \) is on the xy-plane.) (d) \( f(x, y, z) = x + yz, \) and \( C \) is the line segment from \( P = (0, 0, 0) \) to \( (6, 2, 2) \).
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