quiz3-Th-sol

# quiz3-Th-sol - Quiz 3 Thursday(math 32b Problem 1 Using the...

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: Quiz 3 - Thursday (math 32b) November 16, 2007 Problem 1. Using the change of variables (u, v) (x, y) given by x = uv 2 , y = uv, find the area bounded by the curves y 2 = x, y 2 = 4x, x = y and x = 2y. Solution: First, find the equations for the boundary in terms of u and v: y 2 = x u2 v 2 = uv 2 u2 = u u = 1 y 2 = 4x u2 v 2 = 4uv 2 u = 4 x=y x = 2y uv 2 = uv uv 2 = uv v = 1 uv 2 = 2uv v = 2 Now compute the Jacobian: (x, y) = (u, v) The area is given by 2 1 1 4 2 4 v2 v 2uv u = v 2 u - 2uv 2 = -uv 2 . uv 2 dudv = 1 v 2 dv 1 udu = 7 15 v 3 2 u2 4 | | = = 17.5 3 1 2 1 3 2 1 Problem 2. Compute the integral C x2 ds, where C is the upper semicircle x + y 2 = a2 , y 0. Solution: Parametrize the curve by x = a cos , y = a sin , with [0, ]. Then 2 I = 0 a2 cos2 3 0 a2 sin2 + a2 cos2 d = a cos2 d. 0 By properties of sin and cos , cos2 d = 0 sin2 d. 2 ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

quiz3-Th-sol - Quiz 3 Thursday(math 32b Problem 1 Using the...

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

View Full Document
Ask a homework question - tutors are online