This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**Unformatted text preview: **y = g ( t ), with x ( α ) = a and x ( β ) = b , then ± b a y dx = Example 2.1. (problem 31) Use the parametric equations of an ellipse, x = a cos θ , y = b sin θ , ≤ θ ≤ 2 π , to Fnd the area that it encloses. 3. Lengths of Curves Recall we Fnd the length of a curve by ± b a ² 1 + ³ dy dx ´ 2 dx or ± d c ² 1 + ³ dx dy ´ 2 dy If x and y are parameterized by t as x = f ( t ) and y = g ( t ), with α ≤ t ≤ β , and the curve is traversed exactly once as t increases from α to β , then the length is given by Example 3.1. (problem 47) ±ind the length of the curve. x = e t-t , y = 4 e t/ 2 , and-8 ≤ t ≤ 3 . Section 10.2 Calculus with Parametric Curves 4 4. Surface Area Remark 4.1. ds = ± 1 + ( dy dx ) 2 dx = ² 1 + ³ dx dy ´ 2 dy = Example 4.1. (problem 60) Find the area of the surface obtained by rotating about the x-axis. x = 3 t-t 3 , y = 3 t 2 , and ≤ t ≤ 1 ....

View
Full
Document