2220hw10sol

# 2220hw10sol - Math 2220 Problem Set 10 Solutions Spring...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 2220 Problem Set 10 Solutions Spring 2010 Section 6.1 : 2. Calculate the integral of the function f ( x, y, z ) = xyz over the curve c ( t ) = ( t, 2 t, 3 t ), ≤ t ≤ 2. Solution. The integral can be computed as integraldisplay 2 f ( r ( t )) | r ′ ( t ) | dt , where r ( t ) denotes the parametrization of the curve. After substituting the given function f ( x, y, z ) = xyz we get integraldisplay C f ds = integraldisplay 2 ( t )(2 t )(3 t ) | (1 , 2 , 3) | dt = integraldisplay 2 6 √ 14 t 3 dt = 3 2 √ 14 t 4 vextendsingle vextendsingle vextendsingle 2 = 24 √ 14 4. Calculate the integral of f ( x, y, z ) = 3 x + xy + z 3 over the curve c ( t ) = (cos 4 t, sin4 t, 3 t ), ≤ t ≤ 2 π . Solution. The derivative of the parametrization is c ′ ( t ) = (- 4 sin4 t, 4 cos4 t, 3) and its norm is | c ′ ( t ) | = 5. Thus, the integral is integraldisplay C f ds = integraldisplay 2 π (3 cos4 t + cos 4 t sin4 t + 27 t 3 )(5) dt = 5 parenleftbigg 3 4 sin 4 t + 1 8 sin 2 4 t + 27 4 t 4 parenrightbigg vextendsingle vextendsingle vextendsingle 2 π = 540 π 4 8. Calculate integraltext C F · d s for the vector field F ( x, y, z ) = ( x, y,- z ) where C is the curve c ( t ) = ( t, 3 t 2 , 2 t 3 ),- 1 ≤ t ≤ 1. Solution. The derivative of the parametrization is c ′ ( t ) = (1 , 6 t, 6 t 2 ) and the vector field along the curve is F ( c ( t )) = ( t, 3 t 2 ,- 2 t 3 ). Therefore the line integral over the curve can be computed as integraldisplay 1 − 1 ( t, 3 t 2 ,- 2 t 3 ) · (1 , 6 t, 6 t 2 ) dt = integraldisplay 1 − 1 ( t + 18 t 3- 12 t 5 ) dt = t 2 2 + 9 2 t 4- 2 t 6 vextendsingle vextendsingle vextendsingle 1 − 1 = 0 Alternatively, you can start with the line integral integraldisplay C F 1 dx + F 2 dy + F 3 dz = integraldisplay C x dx + y dy- z dz and compute this using ( x, y, z ) = ( t, 3 t 2 , 2 t 3 ), which leads to the same thing....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

2220hw10sol - Math 2220 Problem Set 10 Solutions Spring...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online