Definite Integrals Questions and Answers

Problem 4 Find the area of the region in the first quadrant that is enclosed by the curves x y2 y x 24 and the x axis

X The lower edge of a dam is defined by the parabola y see figure Use a coordinate 25 system with y 0 at the bottom of the dam to determine the total force on the dam Lengths are measured in meters Assume the water level is at the top of the dam OA SO O B Set up the integral that should be used to find the total force on the face of the dam Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers 14 2 dx 9800 14 y 5 y dy ya 25 20 15 Q Q

Problem 2 Evaluate the following integrals Jos a 3 c b 2 0 In x 2 X TU 3 0 dx cos x V1 6 sin x dx sin 8 sec e de cos 8

1 Define f x 2 for x in 0 1 5 f x 0 for x outside of 0 1 5 Find the area between f x and the x axis over the entire real line Answer

tan arccos 5 an arccos Type an integer or a simplified fraction

How do u evaluate the volume enclosed by x 2 y 2 4 and z 2sqrt x 2 y 2 and z 0 using spherical

R be the region bounded by the following curves Use the method of your choice to find the volume of the solid erated when R is revolved about the y axis x y 2x 2 x 3 and x 6 volume of the solid is 108 e an exact answer Save

Let R be the region bounded by y x x 1 and y 0 Use the shell method to find the volume of the solid generated when R is revolved about the line y 3 Set up the integral that gives the volume of the solid using the shell method Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers OA 2n y 3 1 y dy B 0 JO The volume is dx GREED Type an exact answer

K The figure to the right shows the shape and dimension of a small dam Assuming the water level is at the top of the dam and that the shape of the dam is an isosceles triangle find the total force on the face of the dam OA s B 2 27 Set up the integral that should be used to find the total force on the face of the dam Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers 0 dx 2 1000 9 8 27 y Y dy The total force is C 18 m numbol in the math palette as needed Round to the nearest hundredth as needed 27 m

acceleration due to gravity a How much work is required to wind the entire chain onto the cylinder using the winch b How much work is required to wind the chain onto the cylinder if a 40 kg block is attached to the end of the chain orically from a cylinder attached to a winch Assume there is no friction in the system and that the chain has a density of 6 kg m Use 9 8 m s for the 8 9 8 60 y dy 60 a Set up the integral that gives the work required to wind the entire chain onto the cylinder using the winch Use increasing limits of integration 60 arri Type exact answers The amount of work required is 105840 Type an integer or a decimal b Set up the integral that gives the work required to wind the chain onto the cylinder if a 40 kg block is attached to the end of the chain Use increasing limits of integration 9 8 360 6y 40 9 8 dy 0 Type exact answers of 1 The amount of work required if a 40 kg block is attached to the end of the chain is 129384 J desimal Save

Use both the washer method and the shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y x y 25 and x 0 is revolved about the line y 5 O A OB A 5 B 0 x 900 x 5 SCH 0 dy Set up the integral that gives the volume of the solid as a single integral if possible using the shell method Select the correct choice below and fill in any answer boxes within your choice Type exact answers 25 S 2x y 5 y dy 0 SO 0 The volume is dx dx Save

the following integral J5 1 5 1 Inxx dx fax 6 1 In xxx dx Type an exact answer in simplified form

11 214 the following integral 1Aa dp dp Type an exact answer 7 3

shaped like a box is 60 m high with a face that is 70 m wide A strong wind blows directly at the face of the building exerting a pressure of 160 N m at the ground and increasing with height according to P y 160 2y where y is the height above the ground Calculate the total force on the building which is a measure of the resistance that must be included in the design of the building OA S OB SO Find the width function w y for each value of y on the face of the building w y 70 Use integers or decimals for any numbers in the equation Set up the integral that should be used to find the total force on the building Use the base of the building as the origin Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers dx dy Points 0 of 1 STEED points Save

section with a height of 7 m and horizontal sides of width m and 7 m Assume the length of the trough is 14 m See the figure to the right Complete parts a and b below usya who the cross secuona yoy A y 14 Simplify your answer Use integers or fractions for any numbers in the expression Set up the integral that gives the work required to pump the water out of the tank Use increasing limits of integration Select the correct choice below and fill in the answer boas in complete your choice Type exact answers A B dx

Part 1 of 5 JO 0 Type exact answers HW Score 48 8 11 71 of 24 points O Points 0 of 1 For large distances from the surface of Earth the gravitational force is given by F x GMm x R where G 6 7x10 N m kg is the gravitational constant M the mass of Earth m is the mass of the object in the gravitational field R 6 378 x 10 m is the radius of Earth and x20 is the distance above the surface of Earth in maten Cam parts a through d below dx Save 11 a How much work is required to launch a rocket with a mass of 510 kg in a vertical flight path to a height of 2100 km from the earth s surface Set up the integral that gives the wom required

O Points 0 of 1 cross sections that taper linearly from a radius of 6 cm at the top of the glass to a radius of 5 cm at the bottom The glass is 18 cm high and full of orange juice How much work is required to drink all the juice through a straw if your mouth is 4 cm above the top of the glass Use 1000 kg m for the density of orange juice and 0 8 m s for the acceleration due to gravity OA A y x y 0 9 y 0 9 OB A y x 18 y 0 9 18 c A y x OD A y y 0 9 24 points OA S dy Save Set up the integral that gives the work required Use the center of the bottom of the glass as the origin Use increasing limits of integration Select the correct choice below and in the answer boxes to complete your choice Type exact answers

O Points 0 of 1 Use the shell method to find the volume of the solid formed when a hole of radius 1 is drilled symmetrically along the axis of a right circular cone of radius 6 and height OA S dy 6 B 3xx 6 x dx Part 2 of 2 Model the situation on a set of axes by placing the center of the base of the cone at the origin and the cone s axis along the positive y axis Set up the integral that gives the volume of the solid using the shell method Use nonnegative and increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers The volume is Type an exact answer CETTE of 20 points Save

The figure to the right shows the shape and dimension of a small dam Assuming the water level is at the top of the dam and that the shape of the dam is an isosceles triangle find the total force on the face of the dam Part 1 of 3 OA W y y 3 O c w y y ams Find the width function w y for each value of y on the face of the dam Place the origin at the center of the base of the dam with the y axis going in the vertical direction and the going in the horizontal direction Choose the correct answer below O Points 0 of 1 OB W OD wly 3y 24 points

Part 1 of 3 HW Score 48 8 11 71 of 24 points O Points 0 of 1 A diving pool that is 4 m deep and full of water has a viewing window on one of its vertical walls Find the force on a window that is a circle with a radius of 1 5 m tangent to them of the pool Using the center of the window as the origin find the width function w y for each value of y on the face of the window w y

K Use both the washer method and the shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y x y 25 and x 0 is revolved about the line x 6 choice Type exact answers A B AB JO 0 O B 25 6 6 y dy Set up the integral that gives the volume of the solid as a single integral if possible using the shell method Select the correct choice below and fill in any answer boxes within your choice 0 dx 5 2n 6 x 25 x S dy BRECENT dx

A 60 m long chain hangs vertically from a cylinder attached to a winch Assume there is no friction in the system and that the chain has a density of 6 kg m Use 9 8 m s2 for the acceleration due to gravity a Set up the integral that gives the work required to wind the entire chain onto the cylinder using the winch Use increasing limits of integration 60 a How much work is required to wind the entire chain onto the cylinder using the winch b How much work is required to wind the chain onto the cylinder if a 40 kg block is attached to the end of the chain 6 9 8 60 y dy 0 Type exact answers 60 The amount of work required is 105840 J Type an integer or a decimal b Set up the integral that gives the work required to wind the chain onto the cylinder if a 40 kg block is attached to the end of the chain Use increasing limits of integration ugy 40y dy 0 Type exact answers

Points 0 2 of 1 A heavy duty shock absorber is compressed 4 cm from its equilibrium position by a mass of 700 kg How much work is required to compress the shock absorber 8 om from its equilibrium position A mass of 700 kg exerts a force in newtons of 700g where g 9 8 m s 0 08 S 171 500y dy 0 Type exact answers of 24 points Set up the integral that should be used to find the work required to compress the shock absorber 8 cm from its equilibrium position Use decreasing limits of integration Express all displacements in meters W 2744 J Type an integer or a decimal Save

Find the following integral ex ex e x ex ex ex e x dx dx

Evaluate the following integral Include absolute values only when needed dx 9x 7 dx 9x 7 Question 10 7 2 42 In 9x 7 9

Evaluate the following integral 5 5 2x 3 X 1 2x 3 x 1 dx dx Type an exact answer

After the release of radioactive material into the atmosphere from a nuclear power plant in a country in 1982 the hay in that country was contaminated by a radioactive isotope ha the hay to cows when 15 of the radioactive isotope remains how long did the farmers need to wait to use this hay The farmers needed to wait approximately days for it to be safe to feed the hay to the cows Round to one decimal place as needed

1 2x 4 3x

b 4 x dx 5

a 1 dz

Question 1 100 points a S 1 x da b S c 1 1 d e x dx x 9 8x2 2x x 1 dx 3x 2 2x 4x 1dx 2x 4 3x 8 da

If r t cos 2t i sin 2t j 6tk compute j k fr t dt it i j k C where C is a constant vect

3 3 12 6 12 3 f x g x y g x 3 X 1 3 3 12 y f x

Find the size of the permanent endowment in dollars needed to generate an annual 8 000 forever at an interest rate of 4 compounded continuously Round your answer to the nearest whole number

12 x 7x 12

the appropriate description horizonta paraboloid hyperbolic paraboloid cone hyperboloid of one sh Note Make sure that you can sketch each of these surfaces pa 1 x 2y 3z Che horizontal plane non horizontal plane circular cylinder parabolic cylinder ellipsoid elliptic paraboloid hyperbolic paraboloid cone hyperboloid one sheet v 2 z 6x 2y 3 z y 3x 4 z 3 5 3x 6y 2z 4 6 2 6y 2 4 7 z 6x 2y

0 11 Points Use the midpoint rule with the given value of n to approximate the integral Round your answer to four decimal places 24 1 sin sin x dx n 4

rx f x dx lie Enter your answers from smallest to lar Suppose f has absolute minimum value m and absolute maximum value M Between what two values must Which property of integrals allows you to make your conclusion F x dx F x dx F x dx O F x dx F x dx O If f x 0 for a x b then If m f x M for a x b then m b a s 0 f x dx 0 Need Hein Read it f x dx 0 Metal It f x dx M b a

If m f x M for a x b where m is the absolute minimum and M is the absolute maximum of f on the interval a b then b F x Use this property to estimate the value of the integral s m b a smaller value larger value f x dx M ba 5 tan 3x dx

If f x ex 7 0 x 2 find the Riemann sum with n 4 correct to six decimal places taking the sample points to be midpoi MA

Express the integral as a limit of Riemann sums using right endpoints Do not evaluate the limit 1 x x dx lim n 00 n 1 1

The graph of f is shown a b c d e 6 Evaluate each integral by interpreting it in terms of areas 6 21 15 14 0 f x dx 15 5 f x dx 21 h 2 h F x f x dx f x dx 6 If x dx y f x 12 18 24

Click here to watch the video Establish the identity 1 sin 0 1 sin 0 OB O C tan 0 sec 0 Starting with the right side which shows the key steps in establishing the identity O A O D tan 0 sec 0 tan 0 sec 0 H 2 sin 0 2 cos 0 sin 0 2 cos 0 2 2 sin 0 cos 0 2 sin 0 2 cos 0 tan 0 sec 0 tan 0 sec 0 tan 8 sec 0 tan 0 sec 0 2 by workin DOC 1 Cos 0 1 cos 0 sin 0 1 2 1 sin 0 sin 8 1 20 1 cos sin 0 1 1 sin 0 2 sin 0 1 2 1 cos 0 1 sin 0 1 sin 0 1 sin 0 1 sin 0 Sant 1 sin 0 1 sin 0 1 sin 0 1 sin 0

a Estimate the area under the graph of f x 5 x from x 0 to x 4 using four approximating rectangles and righ endpoints Round your answers to four decimal places R

Write the following expression as a single definite integral of the form 3 R x dx x dx x dx f x f x dx F x f x dx

If f x dx 9 1 and f x dx 5 6 find f x dx

A heavy duty shock absorber is compressed 4 cm from its equilibrium position by a mass of 700 kg How much work is required to compress the shock absorber 8 cm from its equilibrium position A mass of 700 kg exerts a force in newtons of 700g where g 9 8 m s Set up the integral that should be used to find the work required to compress the shock absorber 8 cm from its equilibrium position Use decreasing limits of integration Express all displacements in meters 2 8x10 1 715x105 y dy 0 Type exact answers

How much work is required to move an object from x 1 to x 6 measured in meters in the presence of a force in N given by F x Set up the integral that gives the work done 6 3 dx Type exact answers The work is Type an integer or a simplified fraction 32x acting along the x axis

Let R be the region bounded by the following curves Use the shell method to find the volume of the solid generated when R is revolved about the y axis y 4 2x 2x y 0 and x 0 in the first quadrant Set up the integral that gives the volume of the solid using the shell method Use increasing limits of integration Select the correct choice below and fill in the answer boxes to comple your choice Type exact answers OA SO O B dx JO dy

How much work is required to move an object from x 1 to x 6 measured in meters in the presence of a force in N given by F x Set up the integral that gives the work done dx Type exact answers www 3 acting along the x axis