Worksheet # Math 8 Name:. Each region bounded by the following given curves is revolved about the line indicated. Find the volume by any convenient method. a) y, -ais; about -ais. y, ais; about y ais. c) sin, y ais, between and ; about y ais. d) y, y ais; about ais. e) y ; y ;, ; about the y ais. f) y and y ; about the ais. g) The base of a solid is D, y 9 y 6, the arallel cross sections are equilateral triangles erendicular to the y ais. h) The base of a solid is bounded by y and y, the arallel cross sections are isosceles right triangles erendicular with one leg is on the base and erendicular to ais.. a) The region bounded on the left by the grah of the equation y and on the right by the line joining the oints (, ) and (8, ) is revolved about the ais. Calculate the volume of the resulting solid. The larger of the two regions bounded by the circle with equation y and the line through (, ) and (,) is revolved about the ais. Calculate its volume. c) Eress the volume V h of unch in a 6 in. diameter hemisherical bowl as a function of the deth h of the unch. d) A solid has a circular base of radius r. Cross sections of the solid erendicular to the diameter of this base are squares. Calculate the volume of the solid. e) Show that the volume of the oblique cone shown in the following figure is V r h f) Show that the volume of the oblique cylinder shown in the figure is V r h
g) The base of a certain solid is the region enclosed by y, y,, and. Every cross section of the solid taken erendicular to the ais is an isosceles right triangle with its hyotenuse across the base. Find the volume of the solid.. Evaluate the following imroer integrals d a) d - d c) - d) e) - tan d. Determine whether each integral is convergent or divergent. a) d e d) sin d g) e e d / d 8 e e) 6 7 h) 7 d. Find the area of the region bounded by the given curves. a) y, y, right of 6. Find the volume of the following: d d c) / d sin 7 f) i) sin 8 6 d d y, y between = and =. / a) The region bounded by y ; y for is rotated about the ais. / The region bounded by y ln ; for y is rotated about the y ais. 7. Prove the following: a) Show that d converges if and only if Show that. d converges if and only if 8. Find the values of for which each integral converges: a) ln ln 9. a) According to the error bounded formula for Simson s Rule, how many subintervals should you use d to be sure of estimating the value of in absolute value. ln d d by Simson s Rule with an error of no more than A brief calculation shows that if, then the second derivative of f and 8. Base on this, about how many subintervals would you need to estimate lies between d no greater than.. Euler s gamma function ( gamma of ) uses an integral to etend the factorial function from the nonnegative integers to other real values. The formula is t e t dt. with error a) Show that Then aly integration by arts to the integral for to show that. This gives ; ; 6... n n n n! c) Use mathematical induction to verify the equation n n! for every nonnegative integer.
. a) A force of N stretches a linear sring cm. How much work is done in stretching it cm? A conical tank (right circular cone) is m in diameter cross the to and 6 m high. It is filled with water to a level. m from the to. How much work is eended in doing so if the water is umed in rough the bottom? c) A bucket weighing N and holding.m of water is susended by a roe weighing N/m in a well m dee. The bucket is leaking at a rate of.m /s. How much work is necessary to raise it to the to at a stea rate of m/s? d) Calculate the work required to um the water out from the following tank (full of water). e) A vertical dam at the end of a reservoir is in the form of an isosceles traezoid: m across at the surface of the water, 6 m across at the bottom. Given that the reservoir is m dee, calculate the force of the water on the dam. f) The ends of a water trough are semicircular disks with radius ft. Calculate the force of the water on an end given that the trough is full of water. g) Do roblems 7 on on age.. Find the arc length of the following: a) y from to cosh y ; from y to y e e 8 c) y ln from to e) sec h y y from y / to y d) arccos y y y from y / to y f) Find the distance between, and, by using the distance formula. Now find an equation of the line through the given oints and use an arc-length formula to find the same distance.. Find the surface area of the following revolutions. a) y from to rotated about the -ais. y cosh ; from to rotated about the -ais. c) 9 y ; from, to, rotated about the ais. d) y ; from y to y rotated about the y ais. ln e) y ; from to e rotated about the y ais. f) Find the surface area of a shere of radius R. g) The ortion of the circle y to the right of is rotated about the ais, forming a zone of the shere. Find its surface area. h) Find the lateral surface area of a cone of height h and radius r.
. Find the solution of the differential equation that satisfies the given initial condition. a) y tan ; y d y 8 e ; y c) y d ; y d) d y ; y cos e) y ye sin d g) i) d f) y d e y y h) y d d cos y sin cos ; y j) e y d d. Solve the following alication roblems: a) A bank account earns % annual interest comounded continuously. You wish to make ayments out of the account at a rate of $, er year (in a continuous cash flow) for years. i) Write a differential equation describing the balance B f (t ), where t is in years. ii) Find the solution B f (t ) to the differential equation given an initial balance of B in the account. iii) What should the initial balance be such that account has zero balance after recisely years? A rectangular swimming ool has dimensions, in meters, of by by ; hence it has volume 6 cubic meters, or liters. The ool initially contains ure fresh water. At time t minutes, water containing grams/liter of salt is oured into the ool at a rate of 6 liters/minute. The salt water is instantly and totally mied with the fresh water, and the ecess miture is drained out of the bottom of the ool at the same rate (6 liters/minute). Let S (t ) the mass of salt in the ool at time t. i) Write a differential equation for the amount of salt in the ool. ii) Solve the differential equation to find S (t ) iii) What haens to S (t ) as t? c) Suose a brine containing.kg of salt er liter runs into a tank initially filled with L of water containing kg of salt. The brine enters the tank at a rate of L/min. The miture, ket uniform by stirring, is flowing out at the rate of L/Min. i) Find the concentration, in kilogram er liter, of salt in the tank after min. ii) After min, a leak develos in a tank and an additional liter er minutes of miture flows out of the tank. What will the concentration, in kilograms er liter, of salt in the tank minutes after the leak develos? d) $ is ut into a bank account and earns interest continuously at a rate of I er year, and in addition, continuous ayments are made out of the account at a rate of $ a year. Sketch the amount of money in the account as a function of time if the interest rate is (i) % (ii) % (iii) % In each case, you should first find an eression for the amount of money in the account at time t (in years) e) A bank account earns % annual interest comounded continuously. You wish to make ayments out of the account at a rate of $, er year (in a continuous cash flow) for years. i) Write a differential equation describing the balance B f (t ), where t is in years. ii) Find the solution B f (t ) to the differential equation given an initial balance of B in the account. iii) What should the initial balance be such that account has zero balance after recisely years? 6. Find and d d d y of the following arametric equations a) t t ; y t t t sin t ; y t cos t 7. Find the oints on the curve where the tangent is horizontal or vertical. a) t sin t; y cos t cosh t; y sinh t 8. Find an equation of the tangent to the curve at the given oint by two methods: (a) without eliminating the arameter and ( by first eliminating the arameter. t a) y e ; y t ;, tan t ; y sec t ;,
9. Grah the curve of the following olar equations: a) r cos r 7 cos c) r cos( ) d) r sin( ) e) r cos( ) f) r sin. Convert the following olar equations to Cartesian equations a) r csc r sin r cos c) r csc e d) r cos sin e) r r cos f) r sin cos. Find the intersections of the following curves a) r cos and r cos r sin and r sin c) r cos and r cos. Find the horizontal and vertical tangents to the grah of the following: a) r sin for r cos for. Find the area of the following regions: a) Inside the cardioid r a cos for a> Inside one leaf of the four-leaved rose r cos c) Shared by the cardioids r cos and r cos d) Shared by the circle r and the cardioid r cos e) Within the inner loo of r sin f) Inside both r sin and r sin