A RC L ENGTH AND S URFACE A REA Compiled by Mrs. King
S TART WITH SOMETHING EASY The length of the line segment joining points (x 0,y 0 ) and (x 1,y 1 ) is (x 1,y 1 ) (x 0,y 0 ) www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt
T HE L ENGTH OF A P OLYGONAL P ATH ? Add the lengths of the line segments. www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt
T HE 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
A PPROXIMATE THE CURVE WITH POLYGONAL PIECES ? www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt
W HAT 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.web
A RC L ENGTH 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
E XAMPLE P ROBLEM Compute the arc length of the graph of over [0,1].
N OW COMES THE FUN PART … 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!
E XAMPLE Find the arc length of the portion of the curve on the interval [0,1] phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt
Y OU TRY Find the arc length of the portion of the curve on the interval [0,1] phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt
R EVIEW : Find the volume of the solid created by rotating about the x-axis on the interval [0,2] Picture from: http://math12.vln.dreamhosters.com/images/math12.vln.dream hosters.com/2/2d/Basic_cubic_function_graph.gif
S URFACE A REA OF S OLIDS OF R EVOLUTION 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
W HAT 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