Review Problems for the First Exam Integration 1 Improper Integrals 1 Two of your classmates R are having some trouble with improper integrals They are discussing 0 f x dx where f is defined positive and bounded on 0 and limx f x 0 R Todd believes that 0 f x dx ought to diverge He reasons that if f is positive then then the accumulated area keeps increasing even if only by Rb a little bit so we can t get anything other than infinity since 0 f x dx increases with b and we need to let b go to infinity Amani on the other hand is convinced that if limx f x 0 then R f x dx ought to converge After all he reasons the rate at which 0 area is accumulating is going to zero That should be enough assure convergence Todd and Amani ask you for assistance Explain very clearly the errors each of them are making R There is really not enough information given about 0 f x dx to draw any conclusion Illustrate this by providing two integrals of this form satisfying the conditions given one of which converges and one of which diverges R 2 For what values of p does 1 xdxp converge R1 3 For what values of p does 0 xdxp converge R 4 For what values of p does 0 xdxp converge 5 Let A be the region bounded above by the x axis below by the curve y x1 and on the right by the line x 1 a Does the region A have finite area If so what is it b Let V be the volume generated by revolving area A about the x axis V is sometimes called Gabriel s Horn Find the volume of V Curiously it is finite even though A is infinite c Let W be the volume generated by revolving area A about the y axis Is the volume finite If so what is it R 10 6 Does the integral 0 x2dx 3 converge R 3 7 Does the integral 0 e x dx converge R 8 Does 1 sinx x2 converge Hint Analyze by comparison 1 2 Slicing Note for any of these problems you ought to be able to write a Riemann Sum approximating the quantity sought and by taking the limit of the Riemann Sum be able to obtain the integral giving the sought after quantity 9 A beam of light is shining onto a screen creating a disk of radius 50 cm The intensity of light is greatest at the center an diminishes away from the center It the intensity of light at a distance r from the center of the beam is given by f r watts square cm find an expression for the total wattage of the beam s image on the screen 10 At Three Aces pizzeria the chef tosses lots of garlic on the pizza The density of garlic varies with x the distance from the center of the pizza x and is given by g x x3 2 2 ounces per square inch of pizza If the pizza is 14 inches in diameter and Three Aces cuts six slices from each pizza how much garlic is on once slice of pizza 11 A very thin lighted pole 10 feet tall is placed upright in a family s backyard to attract insects to it At one moment the density of these insects is given 13 by r r 1 insects per cubic foot where r measures the number of feet from the pole a How many insects are within 5 feet of the pole at a height of 10 feet or less b How many insects are within 5 feet of the pole at a height of 10 feet or more 3 Volumes 12 Let A be the region bounded above by the y tan x below by the x axis and on the right by x 4 Find an integral giving the volume obtained when the region A is rotated about a the y axis b the line x c the line y 3 You need not evaluate the integrals 13 Consider the region R in the plane bounded by the parabola x y 2 and the line x 9 Now consider an object in three dimensional space with R as its base Its cross sections parallel to the y axis are circles with diameter on the base region What is the volume of the object 2 14 Derive the formula V 34 R3 for the volume of a sphere using the method of slicing 15 Find the volume of the cap of a sphere with radius r and height h See the picture on p 464 of Stewart 23 16 A conical frustum is the region formed by starting with a cone and then chopping it off by a plane parallel to the base Find a formula for the volume of a conical frustum whose inner radius is r outer radius is R and height the distance between the two bases is h 17 Find the volume of the volume obtained by revolving the plane region enclosed by x y 2 and x 2y 3 around the x axis 18 Use the method of washers to find the region obtained by taking the plane region enclosed by y x y x about the line y 1 When you finish find the same problem using the method of shells and check that your answers agree 19 A Wisconsin cheese factory makes its cheese in solid cylinders of radius 2 inches A wedge of cheese is cut from the cylinder by chopping through the diameter of the base at an angle of 45 degrees with the base Find the volume of the wedge of cheese Hint the base of the wedge is a semicircle of radius 2 The cross sections perpendicular to the base are isosceles right triangles 4 Arc Length 20 Find the arclength of R the function f x ln sec x from x 0 to x 4 You may use that sec x ln sec x tan x 5 Average Values 21 Let f x mx b be a linear function on the interval c d Show R d that the mx b dx average value of f is attained at the midpoint f c d 2 c d c R9 22 A continuous function f x satisfies the identity 1 f xx dx 2 What is the average value of f x on the interval from 1 to 3 Your answer should be a number 3 6 Work problems 23 Filled with the desire to be alone you have climbed a tree which is 10 meters high So that you will be able to stay there for a while you have arranged a basket containing lunch and some reading materials weighing 15 kg in all Before climbing up the tree you have attached a sturdy chain to the basket the chain weighs 1 kg per meter If you now pull the picnic basket up to you via the chain how much work do you do 24 Consider …