Definite Integrals Questions and Answers

Q5 b Evaluate the integral fff cos xyz dV V x y z 0 x 1 0 y 1 0 z 1

Express 4 58585858586 as a rational number in the form P 9 where p and q are positive integers with no common factors and a n

A Find the average value of the function over the indicated interval B Graph the function and its average value over the indicated interval in the same viewing windo f x x 1 8 A The average value is 0 Type an integer or decimal rounded to three decimal placor ac pooded

Evaluate the definite integral 4 3e x dx e 3x x 4 3e 3x

7 The rate of growth of a bacteria is being tested in a lab It is found that the bacteria multiplies at a constant rate every hour If the sample started with 5 bacteria and after 4 hours contains 327 680 bacteria determine the rate of growth of the bacteria 1 Point

Determine whether the statement is true or false True False If lim f x exists but lim g x does not exist then lim f x g x does not exist x a x a x a

Use a familiar formula from geometry to find the length of the curve described and then confirm using the definite integral r 6 sin 0 0 0 T Length

Evaluate the integral 34e 80 7e 10 da Answer C Hint let u e so that du e dx and thus dx du u This substitution converts the integrand into a rational function in u which can be integrated after partial fraction decompositi

the area of the region bounded by the x axis and the graph of y sin x from x 0 to x 3 using six rectangles and using right endpoints Estimate Draw the region and show your six rectangles Write your solution on paper and upload an image of a PDF file of your solution

Determine the definite integral f x dx that is defined by the limit tim 1 2 1 lin In 11 i 1 Type your complete solution in the space provided Explain what you did 3i 3 3 21 n n

The function f x 11 sin 32 2 models the average rate of change in experiment In the function x represents the number of minutes since the commenc substance is measured in degrees Fahrenheit Over what range of temperatures does the average rate of change in temperature fall Enter a value in each box to correctly complete the statement The range in the average rate of change in temperature of the substance is from a low F

17 Find the volume of each solid obtained by rotating the bounded region around the given axis using cb the shell method Remember with shell method the general equation is 2 To rh dy for a x axis rotation b and 2 height of the rectangle S rh dx for a y axis rotation where r is the radius usually x if it s dx and y if it s dy and h is the a Region bounded by y 2x x 2 and the x axis Rotated about the y axis

Find the area under the curve 2 a f x x 1 over 1 1 b f x sinx cos x 3 S over 0 1 2

In 66 66 integrate Check Answer Save 6 sin x y dydx if we change the order of integration that is What is True False Step By Step Example sin x y dy Live Help EXT sin x y dxdy then it becomes easier t

Evaluate the integral a b 2 L Cove 1 x 3x dx 2 y 1 y dy 1

Evaluate surface integral where S is surface z x y 0 z 2

Find the volume under the surface z f x y and above the rectangle with the given boundaries z 0 x 1 1 y e 1 2 1 4 1 6 1 3

Given F yi xj Ok and parametric surface 7 u v ui v u v k for 0 u 2 and 0 y 1 find Given 7 u v ui v u v k find ru x F Answer format a b c EX F ds F ds

To show that F Pi Of is a conservative field we need to show that OP dy e sin y Answer format Enter the value Congratulations your answer is correct and saved Check Answer Save Step By Step Example 20 Find dx sin y Answer format Enter the value Congratulations your answer is correct and saved Check Answer Save Step By Step Example Is F a conservative field Check Answer Save Step By Step Example a No b Yes Live Help Live Help 0 Live Help 10 00 dr Find OP dy then find the potential function

x dx xy dy where is the positively oriented curve represented by the circles with radii 1 and 2 centered at the origin Evaluate Find the value of 20 ox OP dy Haec

Evaluate the curve line integral x 2y ds where C C C with C representing the curve y 2x from 0 0 to 1 2 and C is the horizontal line y 2 from 1 2 to 2 2

Find a series expansion centered at x 0 for So t 6 sin t dt 16 Ma

Find the area of the region enclosed by the curves y x 1 and y 7 x

1 point Evaluate the triple integral where E is the solid 0 z 5 0 v z 0 x v xyz dv

Js F e sin y e cos y yz S is the surface of the box bounded by the planes x 3 y 2 z 4 and the coordinate planes

Find the flux of F 2x 7 5z3 2yk out of the closed cone x 2 with 0 x 2 flux

sing disks or washers find the volume of the solid obtained by rotating the region bounded by the curves y x x 3 and y 0 about the z axis olume 108 5 T

Find the flux of F 3x 2zj 3yk out of the closed cone x i y 2 with 0 x 1 flux

2 10 points Suppose that f f x dx 1 and f x dx 13 Find the value of 6f x 1 dx 2x

Find a the area of the region b the volume of the solid when rotated around the x axis and c the volume of the solid when rotated around the y axis Use whichever method seems most appropriate to you if y 5 x y r x 0 and x 1

2 Find the surface area and volume if y is rotated around the x axis from 1 to 4 3 Find the following integrals a

Find the flux of the vector field e 3z xy across the curved sides of the surface S x y z z cos y y 1 0 z 3 Nom vectors point upward

Evaluate the the integral J F ds where F x y xy x and c is the part of the circle x y 1 that starts at 1 0 and ends at 0 oriented counterclockwise

Find a formula to approximate the above integral using n subintervals and using Midpoint Rule enter a formula involving n alone Evalute the formula using the indicated n values 10 100 1000 3x 7 da Find the limit of the formula as noo to find the exact value of the integral

Compute the volume of the solid whose base is the region in the first quadrant bounded by y z y 1 and the y axis and whose cross sections perpendicular to the z axis are squares V 1

20 The length L in ft of wire needed considering the sag between two poles exactly 40 ft apart is L 2 7 1x107x 1 using the trapezoidal rule 0 1 dx With n 10 evaluate L 0 Round the final answer to three decimal places as needed Round all intermediate values to six decimal places as needed

1 NG For the regions bounded by y 6cosx and y x 7 Part A Find the limits of integration 10 points Part B Write an expression representing the total area 10 point Part C Solve for the total area 10 points

ind the volume generated by rotating the region enclosed by y e and y 0 between the interval m4 x In 6 about the line x 1 Simplify your answer as much as possible

a If F x e find F x 2xe x b Find So 1 0 0 8 0 6 0 4 0 2 da in two ways i Using the Fundamental Theorem of Calculus i Numerically using a right Riemann sum with n 5 You can use the graph of 2xe 0 2 0 4 0 6 0 8 1 0 below

an 1 1 for the sequence a find a formula An 23 2 3 16 22 30 9 b find the limit of the sequence An 2ln n In 5n 1

1 23 P 111 L K K LLL KKK 246 Evaluate the following integrals for a pathy that travels from Ti to Ti in the right half plane and also for a pathy from Ti to i in the left half plane d J sinh z dz a zdz b edz c co Solution The integrals are all independent of path and can be evaluated by finding a primitive a b c L co d sir 7 i z dz 6 xx 2x i 5 5 5 5 Ti cos z dz sin z Ledz e dz e ei e i 1 1 0 sin 72 sin i 2 sinh z dz cosh zi cos z dz ei ni e i ni 2i cosh Ti cosh i ei ni e i i 2i 2 e e 2i ei e Ti i e e T e i e i 0

se disks or shells to find the volume of a sphere with radius r

Calculate the integral of ze over the surface of the box with 0 x 2 0 y 3 0 z 6 Express numbers in exact form Use symbolic notation and fractions where needed ze x y ds 1 1

ate 1 sin log y y4 et 2 r sin logy dydr y4 0 H dyd

Find the mean of 12 15 19 23 28 Round your answer to the neareste tenth 040

plane is flying with an airspeed of 180 miles per hour with a heading of 109 The wind currents are a constant 24 miles per hour in the direction of due north Find the true course and ground speed of the plane Round your answers to the nearest whole number X ue course 73 round speed 202 x mph

Confirm the divergence theorem in the region 0 z 10 x y for the oblem 6 ctor field F x y x y 0 by calculating each side of the formula directly

1 point Fill in the following table of values for X I x 0 0 5 f 1 d I x 1 1 5 2

5 Estimating a definite integral and average value from a graph Use the figure below which shows the graph of y f x to answer the following questions Click on the graph to get a larger version A Estimate the integral f f x dx ES You will certainly want to use an enlarged version of the graph to obtain your estimate B Which of the following average values of f is larger o Between x 3 and x 3 o Between 0 and x

5 5 pts 6 6 275 Sketch the region bounded by y region 6 4 2 10 6 4 2 2 ab numbers equations computations or algorithms it is about understanding William Paul Thurston American mathematici 2 6 K 5 x and y 5 Compute the center of mass of