8.1 Integral as Net Change

Transcription

1 8.1 Integral as Net Change Key Concepts/Skill Objectives Chapter 8 Review Guide 1) Solve problems in which a rate is integrated to find the net change over time in a variety of Terms: applications Linear Motion Displacement Total Distance Average Velocity Average Acceleration Work Practice 1) A particle moves along a line so that its position at any time t 0 is given by the function where s is measured in meters and t is measured in seconds. a. Find an equation that can be used to find the particle s rate of change at any time t. b. At what rate is the particle moving at t = 4 seconds? Which direction is the particle moving? c. Find the particle s average rate of change over [0,12]. Indicate proper units. 2) The number of gallons of water in a tank t minutes after the tank started to drain is a. How fast is the water running out at the end of 10 minutes? Indicate units of measure. b. What is the average rate at which the water flows out in the time interval [0,10] 3) (Calculator) A spherical tank contains gallons of water at time t=0 minutes. For the next 6 minutes, water flows out of the tank at a rate of ( ) gallons per minute. a. Find the rate at which water flows out of the tank at t = 4 minutes. b. Write a function W (t) for the amount of water in the tank at any time, t. c. How many gallons of water are in the tank at the end of 6 minutes? 4) The rate at which water is sprayed on a field of vegetables is given by, where t is in minutes and is in gallons per minute. a. Write a function W (t) that can be used to determine the amount of water sprayed on the field at any time, t. b. How much water has been sprayed on the field during the first 10 minutes? 5) The tide removes sand from Sandy Point Beach at a rate modeled by the function R given by. How much sand will have been removed by the tide after 6 hours? 6) The rate at which raw sewage enters a treatment tank is given by gallons per hour for hours. Treated sewage is removed from the tank at a constant rate of 645 gallons per hour. The treatment tank is empty at time t = 0. a. At what rate is the sewage entering the tank when t = 3 hours? b. How many gallons of sewage enter the treatment tank during the time interval? c. How many gallons of sewage are in the tank at t = 3 hours? 8.2 Areas in the Plance Key Concepts/Skill Objectives 1) Use integration to calculate areas of regions in a plane Terms: Area between Curves Integrating with respect to Y Boundaries

2 Practice For each, find the area of the region enclosed by the curves (no calculator): 1) 2) and and 3) and 4) and 5) and 6) and over * Volumes Key Concepts/Skill Objectives 1) Use integration by slices or shells to calculate volumes of solids 2) Use integration to find volume or surface area of a solid from revolution Terms: Volume = Cross-Sections Disk Method Washer Method Practice Cylindrical Shells 1) (Calculator) R is a region bound by the x-axis and the graphs of y = ln x and y = 5 x, as shown in the figure. Region R is the base of a solid. For the solid, each cross-section perpendicular to the x-axis is a square. Write, but do not evaluate, an integral expression that gives the volume of the solid. 2) Region R is bound by the x-axis,, and. Region R is the base of a solid. For each y, where 0, the cross section of the solid taken perpendicular to the y-axis is a rectangle whose base lies in R and whose height is 2y. Find the volume of the solid. 3) Find the volume of the solid of revolution generated by revolving the region bounded by y = x, y = 0, and x = 2 about: (a) the x axis and (b) y axis 4) Find the volume of the solid of revolution generated by revolving the region bounded by y = x 2 and y = 4 about the x axis 5) Find the volume of the solid obtained by rotating the region bounded by the x axis and the graph of y = 1 x 2 about the line y = -3 6) The solid lies between planes perpendicular to the x-axis at and. The cross sections between these planes are circular disks whose diameters run from the curve to the curve. 7) Find the volume of the solid generated by revolving the region bounded by the parabola and the line about a) the x-axis, b) the y-axis, c) the line x = 4, and d) the line y = 4.

3 Answer Key 8.1 1) a) b) c) Moving left because v(4) < 0 2) a) b) 3) a) ( ) ( ) gal/min b) ( ) c) ( ) gallons 4) a) b) gallons 5). * + ( ( )) 6) a) b) ( ) c) ( ) 8.2 1) ( ) * + 2) () * + *+

4 3) ( ) * + 4) x = 0, 2, 4 ( ) ( ) [ ] *+ 5) ( ) * + * + 6) [ ] *+ ( )

5 8.3 1) s = ln x s = 5 x 2) and Boundaries: [0, 2] * + 3) a) about the x-axis b) about the y-axis Boundaries: [0, 2] Boundaries: [0, 2] R= x 0 = x R= 2 y * + * + * + 4) Boundaries: R= 4 0 = 4 r= x 2 0 = x 2 * + *+ * +

6 5) Boundaries: [-1, 1] R= 1 x 2 (-3) = 4 x 2 r= 0 (-3) = 3 * + *+ 6) Boundaries: * + D = 2sinx 2cosx R = ½(2sinx 2cosx) R = sinx cosx * + *+ 7) Boundaries: P(0, 0) to P(4, 4) a) about x-axis b) about y-axis R= 2 x - 0 r= x R= y - 0 r= y 2 /4 0 ( ) c) about x = 4 d) about y = 4 R = 4 x r= 4-2 x R= 4 y 2 /4 r= 4 y ( )