Definite Integrals Questions and Answers

Sketch the region enclosed by the given curves 1 y 16x y X 0 16 10 8 y 10 8 aff 2 Find the area of the region Need Help Read It 10 Watch It 15 8 X X y 10 8 10 6 y 10 2 10 8 15 X X

Find the values of c such that the area of the region bounded by the parabolas y 4x c and y c 4x2 is 288 Enter your answers as a comma separated list C

Find the area of the shaded region 20 15 10 5 y 9x x 2 y 2x 4 6 7 14 00 8 10 X

MY NOTES 6 ASK YOUR TEACHER Sketch the region enclosed by the given curves y 14x1 y x 5 4 2 y 25 20 15 10 5 y 15 10 5 10 Need Help Read It 15 Find the area of the region PRACTICE ANOTHER 2 4 6 X 6 6 4 4 2 y 15 10 5 10 15 y 25 20 15 10 5 2 4 4 6 6 X X

The total accumulated costs C t and revenues R t in thousands of dollars respectively for a photocopying machine satisfy 1 C t 55 and 46 R t 5t5e t6 where t is the time in years Find the useful life of the machine to the nearest year What is the total profit accumulated during the useful life of the machine The useful life of the machine is year s Round to the nearest year as needed F www Using the useful life of the machine rounded to the nearest year the total profit accumulated during the useful life of the machine is Round to the nearest dollar as needed

Calculate the indicated Riemann sum S4 for the function f x 19 3x Partition 0 12 into four subintervals of equal length and for each subinterval XK 1Xk let ck 2xk 1 xk 3 S4 Simplify your answer CIE

2 Find the volume generated by revolving the regions bounded by the given curve about the x axis y x 1 x 0 x 3 y 0

Click here to watch the video Given vectors u and v find a 3u b 3u 5v c v 5u u 4 5 v 2 0 a 3u CND

Calculate the definite integral 1 10 0 x dx

1 Evaluate the definite integral sine sine tan 0 a sec 0 h 2 x 5 dx de

5 The elevation of a path is given by f x x 5x 30 measured in feet where measures horizontal distances in miles Draw a graph of the elevation function and find its average value for 0 x 4

A company determines that its marginal revenue per day is given by R t where R t is the total accumulated revenue in dollars on the fth day The company s marginal cost per day is given by C C where C t is the total accumulated cost in dollars on the fth day R t 90e R 0 0 C t 90 0 8t C O 0 a Find the total profit P T from t 0 to t 10 the first 10 days P T R T C T S R C 0 at dt The total profit is Round to the nearest cent as needed b Find the average daily profit for the first 10 days The average daily profit is Round to the nearest cent as needed

Find the area between the curve and the x axis over the indicated interval y 121 x 11 11 The area under the curve is Simplify your answer

F x sindt sindt F 0

Find the area of the region bounded by the graphs of the given equations y x y x 3333333 A B 1 2 X Mi x dx dx 0 25 0 25 G

Find the area of the region bounded by the graphs of the given equations y 4x y x x 0 x 2 Set up the integral s that will give the area of the region Select the correct choice below and fill in any answer box es to complete the choice OA OB S Dax dx O C fo dx dx

Evaluate the limit lim 3 tan 4x Enter a simplified multiple of or o or 818

dp The marginal price dx at x units of demand per week is proportional to the price p There is no weekly demand at a price of 150 per unit p 0 150 and there is a weekly demand of 7 units at the price of 119 96 per unit p 7 119 96 Use the given information to answer the questions a Find the price demand equation p x Type an exact answer in simplified form Round all decimal values to the nearest hundredth as needed

The marginal cost in dollars per square foot of installing x square feet of kitchen countertop is given by C x x a Find the cost of installing 50 ft2 of countertop b Find the cost of installing an extra 17 ft2 of countertop after 50 ft2 have already been installed a Set up the integral for the cost of installing 50 ft2 of countertop C 50 dx The cost of installing 50 ft of countertop is Round to the nearest cent as needed b Set up the integral for the cost of installing an extra 17 ft2 after 50 ft has already been installed C 50 17 C 50 dx 50 ELDB The cost of installing an extra 17 ft2 of countertop after 50 ft has already been installed is Round to the nearest cent as needed 4

Find s t where s t represents the position function v t represents the velocity function and a t represents the acceleration function a t 6t 4 with v 0 7 and s 0 5 t SOCIED

4 12 pts Based on historical data for population growth rates and the initial number of eagles present in a region biologists expect an eagle population to grow according to the model 200 P t 1 82 33e 0 162t where t is measured in years a How many eagles were initially present b When will the population reach 100 eagles Give exact and approximate solutions

Suppose that f is the function given in the graph below and that f is a piecewise function whose parts are either portions of lines or portions of circles as pictured 6 3 4 3 A 6 2 A 1 1 6 5 4 3 2 1 2 3 5 6 Let A be the function defined by A 2x f t dt b Compute the followings 5 q a What does the Second FTC tell us about the relationship between A and f Write an equation to describe the relationship 0 1 pt 30 29

roblem 3 Evaluate the following integral Problem 3 3 6 cot x In sin x dx Enter yo as in the

roblem 1 Evaluate the following integral 4 x 2 x 5 dx Don t forget the absolute value in your answer Problem 1 Enter your answer as a symbolic function of x as in these examples Don t forget the absolute in your answer Use abs for absolute v

Problem 5 Evaluate the following integral 3 8 1 So 64x 9dx 0

Problem 1 Evaluate the following integral S 12 sin 6x dx

Problem 3 Evaluate the following integral dx 1 x 16 A j 7 16 is arctan C B 16 Tarctan f C 2 16 arctan C 2 2 D 16 1 6 arctan C E 16 1 arctan C F 16 arctan C 16 16 arctan C H 216 arctan C G Problem 3 Select

What is the value of the definite integral exact fraction if necessary Provide your answer below 3 1 2 21 1 2 di Enter your answer as an

1 6 Find the area y 3x 15 and y x 5 and the y axis x20 LOW

T 1 A graphing calculator is recommended Sketch the region enclosed by the given curves 1 x3 y y 1 x2 0 5 0 5 y 1 0 0 5 0 5 1 0 y 1 0 0 5 0 5 0 5 0 5 1 0 1 0 M 0 5 0 5 al y 1 0 0 5 y 1 0F 0 5 1 0 0 5 0 5 JS Secti 0 5 0 5 1 0 1 5 1 0 2 0

Find the area of the shaded region y 3 2 1 1 1 5 10 y x y XIT 15 20 X 27 25 PERM X

Evaluate the definite integral 5 6 v1 3x dx

Suppose parametric equations for the line segment between 7 6 and 4 3 have the form a b If the parametric curve starts at 7 6 when t 0 and ends at 4 3 at t 1 then find a b c and d d X Y a bt c dt

The ellipse x 4 6 1 can be drawn with parametric equations Assume the curve is traced clockwise as the parameter increases If x 4 cos t

veyor measurements found below Find the distance across the lake Round your answer to the nearest tenth Draw a snowflake next to your name for 1 point 85 ft 62 120 ft

Use the Binomial Theorem to expand and simplify the expression x 1 5

A particle is released as part of an experiment Its speed t seconds after release is given by v t 0 6t 12t where v t is in meters per second a How far does the particle travel during the first 6 sec b How far does it travel during the second 6 sec

3 Use symmetry to calculate the following integral 2 points 5 123 31 7 sin x dx 5 123

2 Calculate the following integrals 1 point each 2 2 dx A B TT 2 cos x 3 sin x dx Hospel sol In4 3 c S 172743 3 e x dx C ln2 3

1 Given a function of g x on 1 8 and k x 0 on 1 3 k x dx 10 k x 0 on 3 6 k x dx k x 0 on 6 8 k x dx 5 estimate the following integrals 1 point each c f k x dx A k x dx d Are B Sk x dx D fg k x dx

A 0 60 sin 30x dx 2 points each hint set u 30x Table 4 9 Ti for Quiz 4 21 points 2tniog St 5 bine 2 mt 1 B f 3x lnx dxist hint use integration by parts set x and dy

An automobile company is ready to introduce a new line of cars through a national sales campaign After test marketing the line in a carefully selected city the marketing research department estimates that the sales in millions of dollars will increase at the monthly rate of the following function for t months after the campaign has started S t 22 13e 0 1t Ost 24 A What will be the total sales S t t months after the beginning of the national sales campaign if we assume no sales at the beginning of the campaign s t

The marginal price for a weekly demand of x bottles of shampoo in a drugstore is given by the function shown below Find the price demand equation if the weekly demand is 150 when the price of a bottle of shampoo is 4 What is the weekly demand when the price is 7 50 p x 6 000 3x 50 Find the price demand equation Type an equation

A one meter long bar is heated unevenly with temperature in C at a distance x meters from one end at time t given by H x t 120e 0 15t sin x 0 x 1 On a sheet of paper sketch a graph of H against for t 0 and t 1 Think about why your graphs make sense a Calculate each of H 0 2 t H 0 8 t Be sure that you can say in words what the practical interpretation in terms of temperature of these two partial derivatives is and why each has the sign that it does b Calculate x t Again be sure that you understand why it has the sign that it does and what its interpretation in terms of temperature is

Evaluate the following iterated integral xye dy dx 00 1 2 7 by dy dx 00 Type an exact answer in terms of e

For an average person the rate of change of weight W in pounds with respect to height h in inches is given approximately by the following formula 0 0015h dW dh Find W h if W 60 108 pounds Also find the weight of a person who is 5 feet 7 inches tall W h 0 0005h A person who is 5 feet 7 inches tall weighs about Round to the nearest tenth as needed lb

Use a triple integral to find the volume of the wedge bounded by the parabolic cylinder y x and the planes z 14 y and z 0 The volume of the wedge is Com

Problem 4 Find the area of the region in the first quadrant that is enclosed by the curves x y y x 24 and the x axis Problem 4 2 3 64 8 176 192 64 8 176 192 Just Save 5 Enter your answer symbolically as in these examples Submit Problem 4 for Grading 4

Evaluate x 0 4 0 4 7x y dx dy 7x y dx dy Simplify your answer 7x y dx dy x y

Find Sin t of dv df and d gue sin ax dx 1 50 x Sin t Smada Jinxdbie Cost 51 2 Sin ax da V1 x Coss the anche to at 8r 26 2 1 2 pts Ska du Fa 45 3 bol x 2 1 1500 hor