Calculus II SI Exam 1 review Si Leader: Rosalie Dubberke

Part I Volumes by shapes and revolutions

General Procedure 1) Graph any functions 2) Draw a 3D picture of object 3) Partition the graph (x = 1, x=2, x=k-1, x=k…) 4) Find the volume of one “slice” 5) Sum the volume of all of the slices 6) Send the partition to 0 7) Form the integral 8) Solve the integral if necissary

Problem 1 A solid lies between planes perpendicular to the x-axis at x = -1 and x = 1. The cross-sections perpendicular to the x-axis are circular disks whose diameters run from the parabola y = 𝑥 2 to the parabola y = 2- 𝑥 2 Solve for the volume of the solid

Problem 2 The base of the solid is the disk 𝑥 2 + 𝑦 2 =1 the cross-sections by planes perpendicular to the y-axis between y = -1 and y = 1 are isosceles right triangle with one leg in the disk.

Problem 3 Find the volume of the region in the first quadrant bound by 𝑦=(𝑥−1) (𝑥−3) 2 and the x-axis revolved around the y - axis

Problem 4 Find an equation for (but do not solve) the volume of the region in the first quadrant bound by 𝑦= (𝑥−1) (𝑥−3) 2 and the x-axis revolved around x = -7

Problem 5 Write an equation for (but do not solve) the volume of the area below 𝑦= 3 𝑥 from 0 < x < 8 using: Shell method Washer/disk method

Problem 6 Write an equation for the volume of the area between 𝑦=2 𝑥−1 and 𝑦=𝑥−1 revolved around the line x = 6 using: Shell method Washer/disk method Solve one

Part II Surface area and arc length

General Procedure for arc length 1) Use pythagorians theorem for the length of a line 2) Sum the lengths of each individual line 3) Use the definition of a derivative to substitute for ∆𝑦 4) Pull out the ∆𝑥 5) Change the summation to an integral 6) Solve if necissary

General Procedure for Surface area 1) Use what you know about arc length 2) Multiply the arc length by the radius (we are essentially finding and summing many small circumferences) 3) Change to an integral 4) Solve if necissary

Problem 7 Find the length of the curve 𝑦= 4 2 3 𝑥 3 2 from 0 < x < 1

Problem 8 Find the surface area of the equation 𝑦=2 𝑥 revolved around the x-axis from x = 1 to x = 2

Problem 9 Find the surface area of the equation x = 1 - y revolved around the y-axis from 0 to 1

Part III Work

General Procedure For spring/lifting problems: W = 𝐹 𝑥 𝑑𝑥 For water pump problems: 1) Draw a diagram 2) Partition 3) Find an equation for the work needed to lift one “slice” 4) Sum the work for all slices 5) Write the integral 6) Solve if necissary

Problem 10 A cable that weighs 2lbs/ft is attached to a bucket filled with coal that weighs 800 lbs. The bucket is initially at the bottom of a 500 ft mine shaft. Answer each of the following questions: Determine the amount of work required to lift the bucket to the midpoint of the shaft. Determine the amount of work required to lift the bucket from the midpoint of the shaft to the top. Determine the amount of work required to life the bucket all the way to the top of the shaft.

Problem 11 A cylindrical; reservoir diameter 4 ft and height 6 ft is half full of water weighing 10 𝑙𝑏 𝑓𝑡 3 . Find the work done in emptying the water over the top.

Problem 12 A cylindrical; reservoir diameter 4 ft and height 6 ft is half full of water weighing 10 𝑙𝑏 𝑓𝑡 3 . Write an equation for the work done in emptying the water 3 feet over the top.

Problem 13 A force of 1200 N compresses a spring from its natural length of 18 cm to a length of 16 cm. How much work is done in compressing it from 16 cm to 14 cm?

Problem 14 A leaky bucket weighing 5N is lifted 20 m into the air at a constant speed. The rope weighs 0.08 Nm-1. The bucket starts with 2 N of water and leaks at a constant rate. It finishes draining just as it reaches the top. How much work was done lifting the bucket, the water, and the rope?

Part IV Old Exam Questions

Problem 15

Problem 16