Definite Integrals Questions and Answers

Determine the new integration limits for the following area integral after swapping the order of integration 5 2 3 r dr du r dudr

Natural draft cooling tower are shaped like hyperbola for more efficient cooling of power plants The hyperbola in the tower can be modeled by x 2 25 y 2 256 1 with 150m as its height Find the width of the tower at the top and its narrowest part in the middle

Consider the function f x 3 7x 4 x 2 The absolute maximum value is and this occurs at x The absolute minimum value is and this occurs at x

c The amount a person would have to deposit today present value at a 8 percent interest rate to have 1 000 five years from now Round time value factor to 3 decimal places and final answer to 2 decimal places Present value 735 21 x

8 The PARC Co is considering purchasing a new widget machine The company interest rate MARR is 12 Which if either of the two new widget machines for which annual cost data are available should PARC purchase The following annual cost data are available for the two possible alternatives Data Useful Life Years 6 First Cost Salvage Value Annual Benefit WIDGETS R US WEE WILLY WIDGETS 2 250 000 70 000 675 000 60 000 10 000 M O M O Gradient 2 780 000 85 000 870 000 82 000 8 000 a Make a plot showing the NPW for all options also include on the plot the increment between them b Write out the Choice Table for this situation As this question was originally to be solved by hand comment on what that solution would be

find the mobius transformation that maps i to 0 3 to 6 i infinity to 2i

Find the approximations and S algebra system 5 n 6 12 n 6 12 1 F dx Ta ET M EM and 12 Then compute the corresponding errors S Es E and Eg Round your answers to six decimal places You may wish to use the sum command on

The Bureau of abor Statistics American Time Use Survey www bls gov data showed that the amount of time spent using a computer for leisure varied greatly by age Individuals age 75 and over averaged 0 3 hour 18 minutes per day using a computer for leisure Individuals ago 15 to 19 spend 1 0 hour per day using a computer for leisure If these times follow an exponential distribution find the proportion of each group that spends a Less than 15 minutes per day using a computer for leisure Round your answers to 4 decimal places Age 75 and over Ages 15 to 19 Proportion b More than 2 hours Round your answers to 4 decimal places Age 75 and over Proportion

For the following indefinite integral find the full power series centered at t 0 and the open interval of convergence t f t S 1 18 f t C n 0 The open interval of convergence is Give your answer in interval notation dt

Find the area of the region lying between the loops of r 2 4sine 4 3 277 4 3 3 77 8 3 TT

6 Let A x left begin array cc 2 1 2 x 1 2 end array right in M 2 mathbb R quad x in mathbb R Consider the open ball B 1 A 1 C in M 2 mathbb R mid C A 1 1 Here for B M 2 R we use the Euclidean norm B left b 11 2 b 12 2 b 21 2 b 22 2 right 1 2 i Prove that if A x B A 1 then A x GL 2 R ii Find Is 4 1 4 3 A 2 A 3 2 E GL 2 R

Metric Sy The plastic bottles recycled What is this distance in a meters b kilometers m

Throughout this problem we will consider the improper integral justify your answers and show your work a Explain in your own words why the given integral is improper b Rewrite the improper integral in terms of limits 1 c Evaluated dx or show that it diverges X x 1 dx C

Problem 1 Find the general or the particular solutions for the following ODE s 1 2 y 6y 45y 81 sin 6x 18 cos 6x y 4y 8 sin 2x 12 cos 2x

Find the centroid of the region bounded by the graphs of the functions and y 3 sin x y X K2 1 6 X and touching the origin The centroid is at x y where begin array l bar x bar y end array I

as integram 1 Lady dxdy 1 xy 0 0 is an improper integral and could be defined as the limit of double integrals over the rectangle 0 t 0 t t 1 But if we expand the integrand as a geometric series we can express the integral as the sum of an infinite series Show that int 0 1 int 0 1 frac 1 1 x y d x dy sum n 1 infty frac 1 n 2

7 Sketch the graph of a function f such that both of the following are simultaneously true 10 2 x dx 0 2 50 f x dx 10

How many possible encryption keys are there for a Hill cipher using a 2x2 matrix over Z26 Note A matrix is invertible modulo 26 if and only if it is invertible both modulo 13 and modulo 2 Please show detailed steps oh how you obtained the result

a First sketch the flow line of the vector field Y U 2 9 2 3 3 y 5 2 0 0 y y and b then solve the flow line equation to find the flow line

Use the equation y 1 a Determine whether the parabola will open upward or downward The parabola will open downward The parabola will open upward b Find the equation of the axis of symmetry The axis of symmetry is x 0 Type an equation c Find the vertex The vertex of the parabola is 0 9 Type an ordered pair d Find the y intercept The y intercept is Type an ordered pair

Find the area of the shaded region below Area YA x y 2 y 1 y 1 x ex X

he questions about the following polyn 7x 3 1 7 x 5 1

by the intersection of a cone Plane parallel to its side Write an equation of the parabola with the focus of 3 3 and the directrix of y 1

Show that T u v u v 2uv maps the triangle u v 0 v u 5 to the domain D bounded by a 0 y 0 and y 2500 100x Use T to evaluate S 2 x y dx dy

During optimal conditions the rate of change of the population of a certain organism is proportional to the population at time t in hours At time t 0 hours the population is 300 At time t 24 hours the population is 1000 At what time t is the population 500 A B C D t 24 2 t24 la 500 In 1000 t In

Evaluate each expression using the graphs of y E

We take a random sample of 50 from the STA 13 class last year The total number of students was 419 On the final exam the mean score of the sample was 56 3 and the sample standard deviation was 13 8 Solve the following questions 1 5 Question 2 1 pts What degrees of freedom would you use to construct the confidence interval for the mean score of the whole STA 13 class Type the exact degrees of freedom do not round

Find the exact values of the sine cosine and tangent of the angle 105 60 45 sin 105 No Response cos 105 No Response tan 105 No Response

Calculate the multiple double integral ff of the scalar function f x y in 2 variables x y ov D the region D of the 2 dimensional xy space plane Here f 7 7 6xy D triangle ABC with A 0 1 B 1 0 C 0 1

Find the exact length of the curve y 1 4 2 In x 1 x 6

Find the exact area of the surface obtained by rotating the curve about the x ax x3 1 t 6 2x y 2 SX x 1

4 If the infinite curve y e a 0 is rotated about the z axis find the area of the resulting surface

Find the exact length of the curve 36y x 4 4 x 6 yz

Find the exact length of the curve X 8 1 4y2 1 y 3

46 A steady wind blows a kite due west The kite s height above ground from horizontal position z 0 to 50 Find the distance traveled by the kite z 80 ft is given by y 150 40

A Which trig substitution Or 7 sin 0 0x 7 sec 0 Or 7tan 0 O r 49 sin 0 Or 49 sec 0 OI 49 tan 0 B Determine the sides of the triangle A B C I 49 1 is correct for this integral 0 A B C Which integral do you obtain after substituting for z Simplify as much as poss Note to enter 8 type the word theta do D What is the value of the above integral in terms of Don t forget the C E What is the value of the original integral in terms of z Don t forget the C

Does the inverse relation represent a function If yes list the ordered pairs of th f 1 1 9 9 2 7 7 2

1 Evaluate a c 1 6 5 dx 6 1 X b d 1 2 div

Use the given triangle to evaluate the expression If necessary express the value without a square root in the denominator by rationalizing the denominator 4 COS COS R 4 T COS COS 4 4 2 450 459 Simplify your answer Type an exact answer using radicals as needed Use integers or fractions for any numb the expression Rationalize all denominators

Use the x y coordinates in the given figure to find the value of the trigonometric function at the indicated real number t or state that the expression is undefined CSC 2 3 41 3 1 0 1 2 1 4 9 2 0 1 0 1 713 6 1 0 9 1 1 3 Points 0 of Select the correct choice below and if necessary fill in the answer box to complete your choice O A 1 2 3 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Rationalize the denominator CSC OB The expression is undefined

Graph the solution set of the inequality wher x28

14 58 four red cars and three gray ctures on the graph to show as they counted Blue Gray comparison symbol or 3 69 3 Colors of Cars 29 10 18 would use to measure how much water is in a b easuring cup balance

Write a definite integral that represents the area of the region Do not evaluate the integral y x 5 y2 x 5 5 dx

Use the Trapezoidal Rule to approximate fe da for n 4 Round the result to four decim places Please note the equation is e raised to 3 x O 3 2122 3 3423 O None of the answer choices

The region under the graph of y 7 x x 1 on the interval 1 2 is revolved around x axis Find the volume of the resulting solid O None of the answer choices O 12 5 1 In 7 O 49 3 1 In 9 In 12 98n 3 1 3In 3 6In 2 h

Find the average value of g x 5x e x 2 on 0 5 O 1 e 5 O 0 5 1 e 5 None of the answer choices O 0 5 1 e 25

ore 5 8 Question 8 5 8 answered Given the function f x 7x 5a and the Remainder Theorem f 2 l 11

0 1 x x e abc 3 V X Please enter a complete math problem so we can solve it 5 d f x e log In D 7 8 4 5 sin cos tan cot 9 6 lim dx s o X X I X

whether the following improper integrals are convergent or divergent S dx x 1 dx

For the function f x y x y find the volume under the surface for the domain illustrated below a Write the double integral with X integrated first b Write the double integral with Y integrated first c Evaluate the integral both ways using Mathematica