   Chapter 16.2, Problem 10E

Chapter
Section
Textbook Problem

Evaluate the line integral, where C is the given curve.10. ∫C y2z dS, C is the line segment from (3, 1, 2) to (1, 2, 5)

To determine

To Evaluate: The line integral C(y2z)ds for the line segment from the point (3,1,2) to the point (1,2,5) .

Explanation

Given data:

The given curve C is a line segment from the point (3,1,2) to the point (1,2,5) .

Formula used:

Write the expression to evaluate the line integral for a function f(x,y,z) along the curve C .

Cf(x,y,z)ds=abf(x(t),y(t),z(t))(dxdt)2+(dydt)2+(dzdt)2dt (1)

Here,

a is the lower limit of the curve C and

b is the upper limit of the curve C .

Write the expression to find the parametric equations for a line through the point (x0,y0,z0) and parallel to the direction vector v=a,b,c .

x=x0+at,y=y0+bt,z=z0+ct (2)

Write the expression to find direction vector v=a,b,c for a line segment from the point (x0,y0,z0) to the point (x1,y1,z1) .

a,b,c=x1x0,y1y0,z1z0 (3)

Write the required differential formulae to evaluate the given integral.

ddttn=ntn1

Consider the points (x0,y0,z0) as (3,1,2) and (x1,y1,z1) as (1,2,5) .

Calculation of direction vector:

Substitute 1 for x1 , 2 for y1 , 5 for z1 , 3 for x0 , 1 for y0 , and 2 for z0 in equation (3),

a,b,c=13,21,52=2,1,3

Calculation of parametric equations of the curve:

Substitute 3 for x0 , 1 for y0 , 2 for z0 , (2) for a , 1 for b , 3 for c in equation (2),

x=3+(2)t,y=1+(1)t,z=2+(3)tx=32t,y=1+t,z=2+3t

Consider the limits of scalar parameter t are 0 to 1.

0t1

Find the expression (y2z) as follows

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