Definite Integrals Questions and Answers

1 Daniel had 25 to spend at the fair If the admission to the fair is 4 and the rides cost 1 50 each What is the greatest number of rides Daniel can go on Write an inequality that represents Daniel s situation A B How many rides can Daniel go on Show your work Remember that this should be expressed as an inequality C Graph the solution on the number line 2 2 6 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded on the right 8 by the graph of g y and on the left by the y axis for 2 y 9 about the x axis Round your Y answer to the nearest hundredth position V

Calculate the derivative Use symbolic notation and fractions where needed d dx 7e 2x

Find the horizontal asymptotes of f f t 711 3 6412 8 1 6 Let y f t and give your answer as a comma separated list of equations in terms of the given variables Express numbers in exact form Use symbolic notation and fractions where needed horizontal asymptote s PRETREATE

1 The area of the region between the curves over is y e 4x 0 00 X axis is 5x A 2 The volume of the solid obtained by revolving the above region about y e 9

be a one form on w with and R 7x dx edy a smooth function such that with f x f 0 12 f 1 5 Find the line integral of W along the parametric curve a 0 1 R a t t ef t

The region in the xy plane nounded by the graph of y f x and the lines z a and z b is rotated around the line y 5 to form a solid K Top y f x Line y K a each slice is a circle Which of the following integrals represents the volume of this solid 2 x f x 5 dx f f 2 dz fr f x 25 dr O f x 5 dx

Simplify the expression 2 4x 5r2 52 using factoring and exponential rule

35 The graph of is shown Evaluate each integral by inter preting it in terms of areas a ff f x dx c ff x dx e ff f x dx 2 0 2 b f x dx d ff x dx f ff x dx y f x 4 6 8 X

The expression In x 4 ln 2 ln 2 can be simplified to In w where w

11 14 Use the Midpoint Rule with the given value of n to approxi mate the integral Round the answer to four decimal places 11 fe dx n 6 12 x 1 dx n 5 13 x 8 dx n 5 14 x sin x dx x sin x dx n 4

25 26 Express the integral as a limit of Riemann sums using right endpoints Do not evaluate the limit 25 4 x dx 26 1 dx

Find the values of the six trigonometric functions of 0 If an answer is undefined enter UNDEFINED Function Value Constraint 8 lies in Quadrant III sin 0 cos 8 tan 9 csc 8 sec 8 B cos 8 cot 0 5 13

a A rope is to be hung between two poles 40 feet apart If the rope assumes the shape of the catenary Xx y 10 e 20 e 2 20 20 x 2 compute the length of the rope b Suppose that the function y 9 x is revolved about the Y axis Compute the resulting surface area

Question 1 0 5 points Listen Determine the derivative of f x 3 sin x 9x 9 Of x 3 sin x 9 Of x 3 cos x 9 Of x 3 cos x 9 Of x 3 sin x 9 Question 2 0 5 points Listen

7 A table of values of an increasing function f is shown Us the table to find lower and upper estimates for f f x dx 10 X f x 10 14 18 22 26 2 1 12 6 6 30 3 8

decimal places What does the Riemann sum represent Illustrate with a diagram If f x x 4 0 x 3 evaluate the Riemann sum with n 6 taking the sample points to be midpoints What does the Riemann sum represent Illustrate with a diagram a Evaluate the Riemann sum for f x 1 x 1 x 2 7 A table of values of an increasing function f is shown the table to find lower and upper estimates for f x 30

68 Find the area of the region bounded by the hyperbola 9x 4y 36 and the line x 3

Find the area of the region that lies inside r 1 cos 0 and outside r 2 cos E

Find the derivative by determining the limit of a slope as h approaches 0 f x 9x2 O 18x O 9x 9 O 18x

Find the limit lim m f x Xx S 4 3 0 0 2 1 T 7 1 4 3 2 1 3 T 51 S 7 3 1 1 1 1

Evaluate the following limits Consider looking at the graph of the function to help you understand the situation As necessary enter oo for oo and o0 for 0 a lim 7 I b lim 35x 7 6x 35x 7 6x

For the function x 2 x 2 which of the following statements is TRUE f x 7 1 The function has a single horizontal asymptote y 1 The function has two horizontal asymptotes y 1 and y 1 The function does not have any horizontal asymptotes The function has two vertical asymptotes at x 1 and x 1 The function has a single vertical asymptote x 1

Find the volume formed by rotating about the y axis the region enclosed by I 8y and y with y 0

Which expression is equivalent to O 12 27 2 24 2 12 55296 2 12 177147 6 3 2

Evaluate the integral dt 7 6t 09

The area of the region enclosed by the graphs of y 5x and y 7x is

If V x is the rate of change of the volume of a live band in decibels m and is the distance from the stage in m consider the integral Be sure that you can explain in words the meaning of this integral NOTE You have only THREE chances on this problem What are the units of the integral A decibels m 2 B decibels m s C decibels m 1000 V z dr 0 D decibels s E decibels m F none of the above

Evaluate the definite integral 3 14 0 dx x3 6

1 point Let M be the capped cylindrical surface which is the union of two surfaces a cylinder given by x y 4 0 z 1 and a hemispherical cap defined by x y z 1 2 4 z 2 1 For the vector field F zx 2 y 4y 2 yx 8x z4x compute VXF dS in any way you like M V x F dS

Find each value of 0 in degrees 0 90 and radians 0 0 2 without using a calculato 2 2 a cos 0 b tan 0 1 0 8 degrees radians degrees radians

Construct an appropriate triangle to complete the table 0 90 0 2 Function deg rad cot Function Value 3

Use the Midpoint Rule to approximate with n 6 4 5 2 de dx 1 5

Use the Midpoint Rule to approximate the integral with n 3 Answer 2x 7x d 1

f f x dx 9 S Let Find 2 f x dx 2 f x dx 2 dz 5 f a dz 2 dx h 1 and f9f z 5 dz 0 dx 0 2

VxF dS where F x y z 4yzi 4xzj 4 x y2 zk and S is the part of the paraboloid z x y that lies inside the cylinder x y 1 oriented upward 1 point Use Stokes theorem to evaluate VXF

1 point Use the divergence theorem to find the outward flux of the vector field F x y z 3x i 4y2 j 5z k across the boundary of the rectangular prism 0 x 4 0 y 5 0 2 2

Verify the Divergence Theorem for the vector field and region F 6x 92 8y and the region x y2 1 0 z 9 F ds R div F dV

Verify Stokes Theorem for the given vector field and surface oriented with an upward pointing normal F ez 0 0 the square with vertices 8 0 9 8 8 9 0 8 9 and 0 0 9 ScF ds curl F ds

1 point Suppose V f x y 6y sin xy i 6x sin xy F Vf x y and C is the segment of the parabola y 3x from the point 2 12 to 5 75 Then SF F dr

Let F 2x 2y 2x 2z Use Stokes theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points 1 0 1 0 1 0 and 0 0 1 In particular compute the unit normal vector and the curl of F as well as the value of the integral n the unit normal vector VXF The value of the integral is

x x z ds where C is the line segment from 0 1 2 to 6 3 4 C integral

1 point Let C be the curve which is the union of two line segments the first going from 0 0 to 4 2 and the second going from 4 2 to 8 0 Compute the line integral 40 4dy 2dx

Evaluate the integral by interpreting it in terms of areas 8 64 3x 4 dx

Evaluate the integral by interpreting it in terms of areas 7 6x dx

1 point If F V 2x y2 find F dr where C is the quarter of the circle x y2 16 in the first quadrant oriented counterclockwise JF dr

1 point Suppose V f x y 6y sin xy i 6x sin xy j F Vf x y and C is the segment of the parabola y 3x from the point 2 12 to 5 75 Then F dr

1 point Compute the flux of F 4 x z i 7 4zk through the surface S given by y x z with 0 y 16 x 0 z 0 oriented toward the xz plane flux

1 point Compute the flux of the vector field F 8x y zk through the surface S which is the cone x y2 z with 0 z R oriented downward a Parameterize the cone using cylindrical coordinates write 0 as theta x r 8 y r 8 z r 0 with and r 0 b With this parameterization what is d d c Find the flux of F through S flux

1 point Let C be the positively oriented square with vertices 0 0 2 0 2 2 0 2 Use Green s Theorem to evaluate the line integral 2yxdx 7x ydy