Definite Integrals Questions and Answers

1 point Find an equation of a plane through the point 4 5 2 which is orthogonal to the line x 2 5t y 5 2t z 4 1t in which the coefficient of x is 5 Equation

1 a a X 3 2 1 To find the blue shaded area above we would calculate 1 Where b f x dx area f x 2 b

Use only the fact that 3x 6 x dx 54 and the properties of integrals to evaluate the integrals in parts a through 3x 6 0 d if possible 3 Simplify your answer Type an exact answer using radicals as needed OB The integral cannot be calculated using the given information b Choose the correct answer below and if necessary fill in the answer box to complete your choice A 3 x 6 x 0 A x 6 x dx 18 Simplify your answer Type an exact answer using radicals as needed B The integral cannot be calculated using the given information c Choose the correct answer below and if necessary fill in the answer box to complete your choice 0 6x 6 x dx I 3 Simplify your answer Type an exact answer using radicals as needed B The integral cannot be calculated using the given information

7 Suppose ff x dx 3 ff x dx 5 and g x dx 6 Evaluate the following integrals 2 2 4 2 2 Sg x dx 7 Simplify your answer 7 dx 6 7g x dx 42 Simplify your answer 7 Sig x 2 Simplify your answer g x f x dx 11 7 S199 x 9g x f x dx Simplify your answer

Textbook Videos Using Part I of the Fundamental Theorem of Calculus 14 J C

2 Suppose f x dx f x dx 3 ff x dx 5 and 4 g x dx g x dx 6 7 2 2 7 7 Simplify your answer 7g x dx 2 Simplify your answer 7 fg x g x dx 6 Evaluate the following integrals 2

Graph the following function Then use geometry not Riemann sums to find the area and net area of the region described The region between the graph of y 2x 6 and the x axis from 3 x 9 Select the correct graph of y 2x 6 below A OB The area of the region is square units Type an integer or a simplified fraction LY OC D

x from x Approximate the area under the curve y X 5 using a Right Endpoint approximation with 6 subdivisions 2 to

Write the sum using sigma notation A 1 2 3 4 A B 81 B where n 1

2 4 Suppose 7 7 froxidx 3 frox dx 5 and fix 2 2 7 Si g x dx fg x dx 2 dx 6 Evaluate the following integrals

The figure to the right shows four regions bounded by the graph of y x sinx R R R3 and R4 whose areas are 1 x 1 x 1 and 2 1 respectively Use this information to evaluate the following integral 3x 2 S x 2 3x 2 S 2 x sin x dx x sin x dx Type an exact answer using as needed Area 1 R bjes R y x sinx Area 1 4 Area 1 2ar Area 2 1

Use geometry to evaluate the following integral 7 7 ff x dx 1 f x dx where f x 2x if 1 x 5 12 2x if 5 x 7 f x dx Simplify your answer

Determine the following indefinite integral Check your work by differentiation 1 1 x 5x 11 ax slax s 3 4x dx 3 4x 5x 11 dx

XI 4 1 L 3 CH a 2 2 Where To find the blue shaded area above we would calculate f x dz f x 12 3 area cry area b 4

Given f z Find f 9 sin 3x and f 0 4 and f 0 3

ASSIS 0 x y x y 2 dzdy dx 2 SA So 2 14x x y x y 2 dzdy dx

Suppose F t has the derivative f t shown below and F 0 6 Find values for F 1 and F 8 1 3 2 1 1 2 3 F 1 5 5 F 8 3 4 5 102 7 o

For what value of b does 12 12 72 77 02 LIF o O Ono solution 12 20 2 12

Type oo for oo and oo for co Also type U for U Where is f 0 at z Where is f undefined at I 2 What are the critical values of f at Where does f have relative minimums What are these relative minimum values the relative minimum is occurs at z occurs at z which Where does f have relative maximums What are these relative maximum values the relative maximum is which 2 4X Where does f have absolute minimums What are these absolute minimum values the absolute minimum is at z at I Where does f have absolute maximums What are these absolute maximums values the absolute maximum is Where is f 0 VI in the interval which occurs Where is f 0 Va in the interval which occurs

7 In the following change the cartesian integrals into equivalent polar integrals and then evaluate the polar integrals iii 2 4y x y dydx

Question 6 Consider a function f R R Denote by f the n fold composition of f with itself f n fofo of n times In the examples below we will use the n fold composition for two real valued functions f and g A Let f x x Define the following sequence an n 0 1 if n 0 B Let g x 3 847 1 r Define the following sequence b 0 5 g 0 5 if n 0 if n 0 For each of the sequences describe their behavior as n their asymptotic behavior is there a limit for the sequence itself or are there limits for any subsequence s of it You may need computer simulation to find out this behavior Your answer for both A and B should describe the asymptotic behavior in English and also its formalization using predicates and quantifiers the quantifiers can be over either domain N or R If necessary you can introduce constants to denote the limits in your predicate formulas

1 Let C be a positively oriented simple closed contour around the origin Compute Leve dz EXINA 1 Melons

A manufacturer can produce 4860 cell phones when a dollars is spent on labor and y dollars is spent on 3 capital The equation that relates z and y is 90x y 4860 dy a Find a formula in terms of and y for dz dy dz b Find the value of of dy dr at the point 81 16 Round to 4 decimal places

now that the area in the first quadrant enclosed by the curve x a y b 1 a 0 3 0 is given ab 1 a 1 8 a 3 T 1 a 1 3

F Define the following function N x e dt This function represents the area under the Gaussian normal law and is tabulated and widely used available in mathematical libraries It is known in particular that N 0 et N 0 1 Given a real number a 0 compute the following integral for c 0 after having made an appropriate variable change and express it as a function of N x 1 FL 27 a f exp 12 2 da

1 1 cos x 2x abc 3 f x e log In x Sorry we can t solve this one just yet 7 2 VO 4 dx sin cos tan cot 8 LO 5 2 9 6 lim dx s X I X

vFind an equation of the line that has a slope of 10 and a y intercept of 0 3 Write your answer in the form y mx b

18 Two vectors U and v are given in the form of magnitude and direction magnitude and direction of u v be U Magnitude 9 Direction 220 degrees SW v Magnitude 5 Direction 40 degrees NE a Magnitude 1 67 Direction 50 b Magnitude 4 0 Direction 130 c Magnitude 4 0 Direction 220 d Magnitude 3 064 Direction 12

5 Find a coterminal angle between 0 2m for a b C d 11m 6 5T 117

Select the best answer for the 1 A point on the terminal side of an angle 0 is given Find the exact value of the indicated trigonometric function of 0 5 12 find the sin e a b C d 12 13 5 13 1 12 13 5 2002 x 1 lo bohoq bro obu bons He

2 Which equation matches the graph below x 1 y 2 5 5 a b x 3 4 y 6 9 vectors u c x 3 9 y 2 d x 4 x 4 3 1 bis21 1 x 4 1 y 4 2 2 mot 4 3 2 www y 3 Second 5 6 7 10 1 1 1 2 S boh69 2 obulilam 3 4 6 7 C 8 9 10 100A 40 X

14 The point P x 23 lies on the unit circle What is the value of x in simplest form

1 8 1 0 1 0 Graph of y f t The blue line in the graph above defines a linear function y f t A Suppose that 4 is a point that may vary on the t axis Find the accumulated area function A x f t dt that calculates the net area of the shaded region on the interval 0 2 Hint What is the area of a triangle A x B Find the antiderivative for the function f t that has constant term zero Antiderivative F t t 9

Write a quadratic function whose only zero is 11 h x

Let E r 0 2 Os os Os zs 1 8 25 rs 8 Compute y e x y j3d

he integral 3 17 2 17 2 x 17 x y x y g p q 0 dpd de then a a a b b b3 b 1 0 181 f x y z dzdydx is rewritten in spherical coordinates as

82 If f x g x 7 for 35x55 then f x g x dx A 25 g x dx 7 B 2 g x dx 14 C D 25 25 g x dx 28 8 8 x dx 7 E 1 8 g x dx 14

S 1 2 x dx converges What are all values of p for which A p 1 B p 0 C p 1 1 2 D p 1 F There are no values of p for which this integral converges 1

9 Use referencejangles to find an arbitrary angle for sin 210 a 30 b 120 c 330 d 180

17 Find an angle in which the sin x is the same as the tan 2 3 a b RIN C 7 2 d 11

Solve the following logarithmic equa log x log x 3 1

In er 1 4 3 Let f x a Express f e as a limit of Riemann sums b Find f e dt

A right triangle has a hypotenuse of 5 cm and one side length of 3 cm The angle formed at the vertex of the triangle where these two sides meet is a 36 9 Ob 53 1 O c 90 d 143 1

What is the leading term of the polynomial f x defined below 5 3 x x 6x7 7x 2 6x 2

3 Use the sum and difference formulas to find cos x y when sin x quadrant a b I 44 125 J 25 and cos y Both angles are in the 4th d 124 125

5 10 Pts Change the order of integration for the following integral fff x y dydx 2 4 You must draw a clear and carefully labelled plot of the region of integration R 8x 4

28 x y z 3 Use the elimination method to find the solution for y x 5y 4z 7 2x 3y z 0 A 0 B 1 C 2 D 3

Which is the graph of the system 2x 3y 28 A B C D I

17 Find the value of log2 A 212 112 B C 2 D 2

Suppose you know that terms of A and B 7B O 3 O 7B 6A A B 6A ZA A 7B 2 dz A and f dx 1 32 72 de in x dx B What is the value of