Indefinite Integration Questions and Answers

A trough 7 feet long and 1 foot high is filled with water The vertical cross sections of the trough parallel to the ends are shaped like the graph of y x for x 1 to 2 1 The amount of work required to empty the trough by pumping the water over the top edge is foot pounds Recall that the denisty of water is approximately 62 5 pounds per cubic foot

You MUST show ALL work to get credit 1 Graph and set up the integral to find the volume of the solid created by rotating about x 1 the area completely enclosed by the graphs of the following begin array I f x x 2 e x g x e x end array 2 Find the arclength of Y 9 X 2 x 2

1 5 a Use the table of Maclaurin series to find the first few terms of the Maclaurin series of f x sin x b Use this series to find the first few terms of the series for sin x dx c What is the minimum number of terms of the series found in part b necessary to approximate the integral so that error 10 4 Approximate the integral with that number of terms d Is the approximation in part c too large or too small Why e Use the Alternating series Remainder Theorem to find an upper bound for the error of the approximation found in part c

Evaluate the indefinite integral x2 0 dx 4 8x 4

Evaluate the integral by making the given substitution sec Integral sec 3x tan 3x dx W u 3x

Evaluate the definite integral L 2 5 22 28 3 cos z a

Evaluate the indefinite integral COS T 2

Evaluate the indefinite integral cos m

Evaluate the indefinite integral 6 6x 5 5 dx

Evaluate the indefinite integral se sec x tan x 3 sec x 5 2 da

Evaluate the indefinit integral 5 sin 5 sin cos z dx

Evaluate the indefinit integral 1x 3 1x 6x 4 4 dx

2 True or False If f x 0 for all real X values then lim f x x X Show a graph to illustrate you answer

Evaluate the integrals tanh x sech xdx Stan

Find an equation of the line that satisfies the given conditions Through 1 3 perpendicular to the line 2x 7y 3 0 7x 3y 3 0

Evaluate the integral 3 2 1 31 x2 dx

valuate the integral Use C for the constant of integration dx 13 V5x 1 5x

valuate the integral Use C for the constant of integratio e e 5 dx

valuate the integral Use C for the constant of integratio c 2 e dx

financial services committee had 60 members of which 9 were women If 8 members a e probability that the group of 8 would be composed as the following 5 men and 3 women 6 men and 2 women at least one woman The probability that the group will consist of 5 men and 3 women is Round to four decimal places as needed

xo yo such that the initial value problem y x y 1 y x Yo has a unique solution

The regionbounded by the lines a and x b the graph of y f x and the x axis is rotated around the x axis to form a solid 111e each slice is a circle of f z dz of 2mz f 2 dz of f x da of f 2 y f x Which of the following integrals represents the volume of this solid f x da b X

2x z 1 and y A solid is formed by rotating the region in the xy plane bounded by the curves y about the x axis The volume of this solid is

Find or approximate the point s at which the given function equals its average value on the given interval f x 1 0 a where a is a positive real number The point s at which f x equals its average value is are Type an exact answer using radicals as needed Use a comma to separate answers as needed

T I The height of an arch above the ground is given by the function y 6 cos x for 2 sxs What is the average height of the arch above the ground The average height is Type an exact answer using www

Find the average value of the following function over the given interval Draw a graph of the function and indicate the average value f x x x 1 1 6

8 points Use a change of variables to evaluate the following definite integral 4x 2x 1 dx a 1 pt Determine a change of variables from to Choose the correct answer below for A U 4xdx B u 2x 1 C u 2x D 2x 1 U b 1 pt Find du du c 3 pts Write the integral in terms of with new integration limits 4x 1 dx du 2x 1 d 3 pts Evaluate the integral

NAME Objective In this lesson you will learn how to find arithr Find a f g x b f g x c fg x and d f g 1 f x 2x 5 3 f x x g x 2 x g x 4x 5 2 4

Find the quotient Do not round 3 305 100

Find the quotient Round to 3 decimal places 35 5

Round the decimal to the nearest tenth 0 52

Evaluate the integral 9x 6x 6 x dx

valuate the integral Use C for any needed constant 12 3 2 cos x 6 sina

The region enclosed by y 5 y e 0 and x 1 has area e

Problem 8 Find the indefinite integral or antiderivative 1 cos 20 do

Use Stokes Theorem to evaluate VXF dS where M is the hemisphere x y 9 x 0 with the normal in the direction of the positive x direction and F x9 0 M Begin by writing down the standard parametrization of OM as a function of the angle 8 denoted by t in your answer X z y SOM F ds f 0 do where f 0 The value of the integral is use t for theta

al3x ds where C is the right half of the circle x y 25 the line integral

Show that u v 3u 4 u v 5u v parametrizes the plane 2x y z 8 Then a Calculate T T and n u v b Find the area of S D where D u v 0 u 2 0 v 3 c Express f x y z yz in terms of u and u and evaluate ff f x y z ds a T Tu b Area S c f f x y z dS To 3 n u v

Calculate f x y z dS For Part of the surface x z where 0 x y 2 f x y z x Is f x y z ds

1 point Let C be the positively oriented circle x y2 1 Use Green s Theorem to evaluate the line integral 18y dx 8x dy

1 point Find the volume of the region bounded by z x 0 x 6 and the planes y 0 y 6 and z 0 volume 4665 6

Noting that cos bx Re eibe evaluate the integrals a b Jeaz eax cos bx dx Ice ax xe cos bx dx

Suppose the interval 5 7 is partitioned into n 4 subintervals What is the subinterval length Ax List the grid points X0 X1 X2 X3 X4 Which points are used for the left right and midpoint Riemann sums Ax 7 2 1 List the grid points Simplify your answers Use a comma to separate answers as needed

18 Determine each antiderivative a fin 2r 3 dr b fe 2 2x 1 dz In 2x 3 x 2x 3 In 12x 31 C x ex 2xe ex 2x 2 ex 2e c 2 dr dz 1 z 1 x 2 x 3 x 2x 6 m 1xX 21 C In 1x 11 In 1x 21 In 1x 31

14 Let F be an anitderivative for f and G be an antiderivative for g Fill in each blank with the correct antiderivative fkdx kx c 1 fainz dz cos x c X b fr dr n t c In 1x1 C d fc f x dx c fb x C e 2 2 d f x x gx C C k escr cot z dr CSC X c n 1 1 tanz dr In 1sec x C m cot z dz In sin x 1 b e see sec z tanz cos 2 dz sec 2 2 C d 1 1 y dy In lyl arctan y tarcsin y C 2 cos rdr Sin X C h sec zdr tan x C 1 dir cot x C 1 fuertan dr sec x C q dr Jver arcsin x c n secz dz In sec X tan x s 1 cosh cosh z dr 0 fesex dr In lcsc x cot x C P fe dr x C 15 Evaluate each antiderivative a 2 4 dr 4 sin 4 sin w 2 cos w du 4 cos w 2 sin w C dr Jid arctan x c dr arcsec x C sinh x C 1 sinh zdr cosh x C 0 4 3 c 3 t dt cos 30 1 5 2 sin 30 de 5 2sin 38 C 8 2 1 x c h 1 ds In Is in s 1 C slns st dr arcsec

uppose the interval 2 0 is partitioned into n 4 subintervals What is the subinterval length Ax List the grid points x 1x 112 List the grid points 3 Simplify your answers Use a comma to separate answers as needed Which points are used for the left Riemann sum 0 Lise a comm needed Which points are used for the left right and midpoint

Find the average value fave of f x x between 1 and 1 then find a number c in 1 1 where f c fave fave C

x x7 9 7 This can be transformed into a basic integral by letting Consider the indefinite integral 7 U x 9 or and du 7x6 dx dx Performing the substitution yields the integral

felle a This can be transformed into a basic integral by letting Consider the indefinite integral U and 6x 4 dx

Evaluate the integral dx I 6x 2 by making the appropriate substitution u dx 6x 2