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# Class Note for MATH 250A with Professor Lega at UA

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This 10 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15

Calculus and Differential Equations ll MATH 250 B Area volume arc length density and center of mass Areas and volumes 0 Example 1 We want to calculate the area of the region between the curves of equation y X and y for X 6 01 0 Draw a picture and calculate the area using 0 Horizontal slices 9 Vertical slices 9 Select your answer 0 12 a 13 9 16 9 18 9 Example 2 Find the volume of a prism with sides of length 27 3 and 4 centimeters 0 Recall that if you have to choose how to slice an object try to make your task as simple as possible Areas and volumes continued Which of the graphs below represent the area as a function of X of the region between the two curves shown in the plot on the left a Area volume arc length density and center of mass Calculus and Differential Equations l Areas and volumes continued Which of the graphs below represent the area as a function of X of the region between the two curves shown in the plot on the left M in 7r2 7r 37r2 2n 1r2 7r 3M2 2n wk IQ 2 W 312 27 7r2 739 37r2 2 Area volume arc length density and center of mass Calculus and Differential Equations ll Volumes 0 Example 1 Find the volume of the object obtained by rotating the region bounded by y X 1 and y 0 about the axis of equation X 1 0 Example 2 Find the volume of the object obtained by rotating the region R bounded by y expX X 0 X 1 and the X axis about the line of equation y 7 0 Example 3 Find the volume of the object whose base is the region R defined above and whose cross sections perpendicular to the X axis are squares a Which formula represents the volume of the solid obtained by rotating the region between the 2 curves about the y axis o 0q27rxfxegx dx 9 Oqfxegx dx 9 0q7rfxegx2 dx 0 Oq7rfx2e7rgx2 dx 9 quxfxigx dX Area volume arr length density and Center of malt Calculus and Differential Equations Arc length 9 To compute the length of a curve think of a particle moving along the curve and integrate its velocity as a function of time 9 Of course the above assumes that the particle always moves in the same direction along the curve ie that its speed does not change sign 0 Then if the curve is given as the graph of a function say y fx for X E 3 b then its length I is Iab l l f x2 dx 0 Alternatively if the curve is given in parametric form ie if we know Xt and yt for t 6 17 t2 then x t2 y t2 dr Arc length continued 0 Example 1 Consider a curve described by X t and y ft Do the two formulas given above match 9 Example 2 Find the arc length of the curve of equation y X32 for X 6 02 and indicate which of the possible answers below is correct WY11gt WV11gt Was11gt Arc length continued Which of the graphs below represents the arc length of the curve shown on the left a 1 b y 10 e 16 12 12 y 1 2 8 8 10 4 4 8 41iiiu T 41iiiu I 1 2 3 4 J 1 2 3 4 5 6 c y d y 16 e 16 e 4 12 12 2 8 3 I 1 2 3 4 3 4 4 41iiiu r 41iiiu I 1 2 3 4 5 1 2 3 4 a Area volume arc length density and center of mass Calculus and Differential Equations l Density and center of mass 0 Here we only consider objects that are one dimensional As a consequence their density is given in units of mass per unit length eg gcm 9 Example Find the mass of a rod of length 10 cm and of density 6X eXp7X grams per centimeter for X 6 010 0 The center of mass of a collection of N particles of mass m at positions X is the point with coordinate o By analogy the center of mass of a rod of density 6X starting at X a and ending at X b has coordinate fabX6X dX fab 6X dX 39 7

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