Definite Integrals Questions and Answers

Test the series for convergence or divergence 00 n 2 In n n Extra Credit 3 points if you verify convergence divergence using two different methods

INT 3 x 1 20 y 2x 16 87 Find the area shown below Hint The indefinite integral of y 3 is y x C 10 x 4 y 3x x 2

Find the average value of the function f x y z xyz on the unit sphere in the first octant Set up the surface integral of f over the first octant of the unit sphere Use increasing limits of integration SS JSO dy dx 00 Type an exact answer using radicals as needed

Use NINT on a calculator to find the numerical integral of the function over the specified interval f x 12 e X The numerical integral of the function is approximately Round to six decimal places as needed

Complete the table to show the approximate area of the region using n number of rectangles of equal width f x 1 x 2 3 y 6 4 2 4 6 X i

3 Given a Velocity function of v t motion from time t 0 to time t 2 find a Net Displacement of the particle b Total distance traveled for the particle 1 et 2 in meters per second for a particle in 2

A heavy rope 60 ft long weighs 0 7 lb ft and hangs over the edge of a building 150 ft high Let x be the distance in feet below the top of the building Enter x as x a How much work W is done in pulling the rope to the top of the building Show how to approximate the required work by a Riemann sum 6x 10 lim n 8 6 Express the work as an integral 60 6x 1 Evaluate the integral 1260 lim 816 1 1 046 10 b How much work W is done in pulling half the rope to the top of the building Show how to approximate the required work by a Riemann sum x 2 60x ft lb Ax Evaluate the integral Express the work as an integral 30 60x ft lb x dx Ax x 2 x dx

Area between Curves Graph them first 1 If R is a region bounded above by the graph of the function f x by the graph of the function g x x find the area of region R 2 Find the Area between the curves of f x x x and g x x 3 Find the Area Between the curves of r 24 and r 24 and r 1 21

For n since f x dx lim 510 13 is continuous positive and decreasing on 1 o we consider the following If the quantity diverges enter DIVERGES x10

Use the integral test to determine whether the series is 00 n 1 7 n 1 Evaluate the following integral X x 1 Since the integral is not dx finita th

1 point The instantaneous rate of change of the value of a certain investment P is proportional to its value That is to say rP If r 6 and P 0 3500 R A

1 point Solve the separable differential equation Subject to the initial condition y 0 9 y 11x6yvx 1 dy da 0

MY NOTES Calculate the left Riemann sum for the given function over the given interval using the given value of n When rounding round your answer to four decimal places If using the tabular method values of the function in the table should be accurate to at least five decimal places HINT See Example 2 f x 124x31 over 0 2 n 4

Use a left sum and a right sum with rectangles of equal width for the given value of n to approximate the integral Round the answers to two decimal places 22 5x 1 dx n 5 left sum right sum

Determine the area that is bounded by the graphs of the following equations Round your answer to three decimal places y 2x y 7x x

Evaluate the integral Assume that A is a real number and that A 0 0 L 3 x 3 3 dx

4 Q 5 Determine whether the improper integral dxdiverges or converges Evaluate the integral if it converges 8 points

On a sketch of y e represent the left Riemann sum with n 2 approximatinge dx Write out the terms of the sum but do not evaluate it Sum

and estimate the error in using it as an approximation to th Find the partial sum s of the series m sum of the series Round to five decimal places where necessary

5 Let S be the surface obtained by rotating the curve r cos u z sin 2 It around the z axis Use the divergence theorem and an appropriate vector field F to find th volume of S D

For the following exercises find the volume generated when the region between the curves is rotated around the given axis 140 y 3x y 0 x 0 and x 2 rotated around the y axis 141 y x x 0 and y 8 rotated around the y axis 142 y x y x rotated around the y axis 143 y x y 0 and x 1 rotated around the line x 2 144 y 4 145 y x 1 x 2 and y 0 rotated around the line x 4 and y x rotated around the y axis 146 y and y x rotated around the line x 2 147 x y x 1 x 1 and x 2 rotated around the x axis

The region R of the ry plane is bounded by the graphs of y x 1 and y 1 2 and the lines z 1 and x 4 A solid S is generated by revolving S about the line x 4 The volume of S is given by the integral dr

cos x and y 1 2x 1 Find the area of the region between the curves y T Area between curves 1 2

3 x is y x C 15 Find the area shown below Hint The indefinite integral of y 2 87 3 x 1 PL y 4 x 4

Find the area of the region between the curves y cos x and y 1 2x n

Find the area of the region between the curves y x x and y 3x Area between curves

Find the area of the shaded region below Area YA y 5r 12 y x 4 4 x

Find the area of the region enclosed by the two functions y 6x and y x 5 1 0 1 0

Find the most general antiderivative of the function f z z 7 1 22 1 2 where 1 z 1 Use an upper case C for any arbitrary constants F x 4

x Find the area of the region between the curves y x and y x Area between curves

Find the exact area of the region that lies beneath the curve y 7 sin x 0 x T

Find f s if we know that 4 Type your complete solution in the space provided For the toolbar press ALT F10 PC or ALT FN F10 Mac f s 3s 25 1 1 s A and f 1 5

Find f x if we know that Type your complete solution in the space provided For the toolbar press ALT F10 PC or ALT FN F10 Mac f x 4 18 and f 1 9 and f 2 39 4

Work on all of the following problems Evaluate 1 S sec v e tan v dv

Graph each function State the Period domain and rang y tanx 2 Y HE

Use a series to estimate the following integral s value with an error of magnitude less than 10 8 0 3 Sex 0 0 3 e x dx dx

Let S be the surface area which is obtained by rotating the curve 16 22 in the first region about y axis If the surface area S is equal to B3 2 1 then B 9 St streve

Use the Midpoint Rule with n 5 to estimate the volume V obtained by rotating about the y axis the region under the curve y 4 7x 0 sxs 1 Round your answer to two decimal place V

1 Write each sum using summation notation assuming the suggested pattern continues a 2 5 8 11 29 b 1 8 27 n 1 c 5 15 45 135

The income distribution for country A is estimated by the function f x 0 2x 0 42x 0 38 The income distribution for country B is estimated by the function f x 0 64x 0 41x 0 77 2 Find the coefficient of inequality for each of the two countries Round your answers to three decimal places

Consider the definite integral 10 5x dz a Approximate the definite integral with the Trapezoidal Rule and n 4 c Find the exact value of the integral 2440 3 b Approximate the definite integral with Simpson s Rule and n 4 help numb

For the following exercises draw the region bounded by the curves Then find the volume when the region is rotated around the y axis 82 y 4 x x 0 and y 0 83 y 2x x 0 x 1 and y 0 84 y 3x x 0 and y 3 85 y 4x y 0 and x 0 0 and x 3 86 y 87 x 1 1 sec y and y 4 y 0 and x 0 1 x 0 and x 2 88 y 89 y 4 x y x and x 0

3 The graph of a function f t is shown below Y n 1 2 3 4 5 6 7 4 3 2 1 The following questions are above the function A z f t dt a What is A 3 b What is A 5 8 c At which values of x if any does A z have a local maximum 9 10

e Let F x be an antiderivative of f x and let G x be an antiderivative of g x If f x is always less than g x then F x is always less than G x TRUE FALSE

f If f x is a continuous function then F x f t dt is also a continuous function TRUE FALSE

As on the previous page the graph of the function f t is shown below 6 Y 4 3 2 1 1 1 2 3 4 5 6 8 9 10 e Find the absolute maximum value and the absolute minimum value of the function A z f t dt on the interval 0 10

d Splitting an integral along its interval as in n f x dx f ar dx f x dx is only a TRUE FALSE valid if a c b in other words c is between a and b

Step 2 of 2 If we ignore air resistance a falling body will fall 1612 feet in t seconds What is the average velocity between f 9 and t 9 2 Round your answer to two decimal places if necessary Answer 2 Points Keyboard

16 9x 27 x A 2 3 B 1 C 3 3 D IN

While taking a walk along the road where you live you accidentally drop your glove but you don t know where The probability density p x having dropped the glove x kilometers from home along the road is p x 9e for x 0 a What is the probability that you dropped it within 1 kilometer of home b At what distance y from home is the probability that you dropped it within y km of home equal to 0 95 km