Report

ARC LENGTH AND SURFACE AREA Compiled by Mrs. King START WITH SOMETHING EASY The length of the line segment joining points (x0,y0) and (x1,y1) is ( x 0 x1 ) ( y 0 y 1 ) 2 2 (x1,y1) (x0,y0) www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt THE LENGTH OF A POLYGONAL PATH? Add the lengths of the line segments. www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt THE LENGTH OF A CURVE? Approximate by chopping it into polygonal pieces and adding up the lengths of the pieces www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt APPROXIMATE THE CURVE WITH POLYGONAL PIECES? www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt WHAT ARE WE DOING? In essence, we are subdividing an arc into infinitely many line segments and calculating the sum of the lengths of these line segments. For a demonstration, let’s visit the web. THE FORMULA: L b a 1 f ' x dx 2 ARC LENGTH Note: Many of these integrals cannot be evaluated with techniques we know. We should use a calculator to find these integrals. phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt EXAMPLE PROBLEM Compute the arc length of the graph of over [0,1]. f x x L 1 0 3 2 1 f ' x dx 2 3x 1 2 2 1 L 1 0 2 dx 3x 1 2 2 1 NOW COMES THE FUN PART… L 1 0 2 dx First, press the Math button and select choice 9:fnInt( Next, type the function, followed by X, the lower bound, and the upper bound. Press Enter and you get the decimal approximation of the integral! L fnInt ( 1 3 / 2 X 1 / 2 , X , 0 ,1 L 1 . 44 2 EXAMPLE Find the arc length of the portion of the curve 2 on the interval [0,1] y x phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt YOU TRY Find the arc length of the portion of the curve 4 y x on the interval [0,1] phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt SURFACE AREA Compiled by Mrs. King REVIEW: Find the volume of the solid created by rotating 3 f ( x ) x about the x-axis on the interval [0,2] x dx 2 0 2 2 3 x dx 6 0 2 1 7 128 x 7 0 7 Picture from: http://math12.vln.dreamhosters.com/images/math12.vln.dream hosters.com/2/2d/Basic_cubic_function_graph.gif SURFACE AREA OF SOLIDS OF REVOLUTION When we talk about the surface area of a solid of revolution, these solids only consist of what is being revolved. For example, if the solid was a can of soup, the surface area would only include the soup can label (not the top or bottom of the can) phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt WHAT ARE WE DOING? Instead of calculating the volume of the rotated surface, we are now going to calculate the surface area of the solid of revolution THE FORMULA: L 2 b a f x 1 f ' x dx 2 EX 2.5 Find the surface area of the surface generated by 4 revolving y x , 0 x 1, about the x-axis y' 4 x S 2 1 x 0 2 fnInt X ^ 4 4 3 1 4x dx 3 2 1 4 X ^ 3 , X ,0 ,1 S 3 . 437 phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt 2 CLOSURE Hand in: Find the surface area of the solid created 2 by revolving y x , 0 x 1 about the xaxis y' 2x S 2 1 x 0 2 fnInt X 2 2 1 2 x dx 2 1 2 X , X ,0 ,1 S 3 . 81 phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt 2 HOMEWORK Page #