Explanation If C is given by the parametric equations x = x t , y = y t , z = z t , a ≤ t ≤ b Divide the parameter interval a , b into n subintervals t i - 1 , t i of equal width. The i t h subinterval corresponds to a sub arc of C with length ∆ s i . Then ∫ C □ f x , y d s = lim n → ∞ ⁡ ∑ i = 1 n f x i * , y i * ∆ s i Where x i * , y i * is any sample point in the i t h sub arc (b) To explain: How to evaluate the line integral If f is defined on a smooth curve C , then line integral of f along C is ∫ C □ f ( x , y ) d s = ∫ a b f x t , y t d x d t 2 + d y d t 2 d t . Similarly, if f is defined on a smooth space curve C , thenline integral of f along C is ∫ C □ f ( x , y , z ) d s = ∫ a b f x t , y t , z t d x d t 2 + d y d t 2 + d z d t 2 d t (c) To write: The expressions for the mass and center of mass of a thin wire shaped like a curve C if the wire has linear density function ρ x , y . The expression for mass is m = ∫ C □ ρ x , y d s The expression for center of mass is x - , y - , where x - = 1 m ∫ C □ x ρ x , y d s y - = 1 m ∫ C □ y ρ x , y d s (d) To write: The definitions of the line integrals along C of a scalar function f withrespect to x , y , and z

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Chapter 16.R, Problem 3CC

To determine

(a)

To write:

The definition of the line integral of a scalar function f along a smooth curve C with respect to arc length

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Chapter 16.R, Problem 2CC

Chapter 16.R, Problem 4CC

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Consider the line integral I = rydy-y²dr where is the square cut from the first quadrant by the lines a 1 and y = 1. (i) Find the line integral (ii) Find the line integral using the circulation form of Green's Theorem. using the flux form of Green's Theorem.

Suppose the solid W in the figure is the spherical half-shell consisting of the points above the xy-plane that are between concentric spheres centered at the origin of radii 4 cm and 10 cm. Suppose the density 8 of the material increases linearly with the distance from the origin, and that at the inner surface the density is 8 g/cm³ while at the outer surface it is 10 g/cm. (a) Using spherical coordinates, write d as a function of p. Enter p as rho. 8(e) = 25/9(rho-4) (5.0 (b) Set up the integral to calculate the mass of the shell in the form below. If necessary, enter o as phi, and 0 as theta. B D CLI 25/9(rho-4)rho^2sinphi "OP Ópdp A = 0 B = 2pi C = 0 D= pi/2 (Drag to rotate) E- 4 F= 10 (c) Find the mass of the shell. 4536pi

(a) Express the mass M of a thin wire in 3-space as a line integral. (b) Express the length of a curve as a line integral

Chapter 16 Solutions

UD CALC (241 ONLY) W/1 TERM ACCESS >IB

Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

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(a) To write: The definition of the line integral of a scalar function f along a smooth curve C with respect to arc length

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(a) To write: The definition of the line integral of a scalar function f along a smooth curve C with respect to arc length

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(a)

To write:

The definition of the line integral of a scalar function f along a smooth curve C with respect to arc length