Definite Integrals Questions and Answers

Evaluate the double integral over the given region R S Sex YdA R ex YdA R 0 x In 2 0 ys In 5 Choose two integrals that are equivalent to DA C In 5 In 2 0 S Sex ye R 0 In 5 In 2 S S exy dx dy 0 0 dA ex y dy dx Ssex s R www dA Type an integer or a simplified fraction B In 2 In 5 S S ex Y dy 0 0 In 2 In 5 S 0 0 Sex y dx c

3 Use Simpson s Rule in Maple with n 32 to approximate the integral Vsin x dx Take a picture of your Maple output and upload it to Moodle as a pdf together with your worksheet solutions

0 Evaluate the definite integral action 5 12 Cinx dx x

9 Use the definite integral to find the area between the x axis and FCX over the indicated interval Check firs to see if the graph crosses the x axis interval in the given F x 4 1 5 The area between the x axis and F x is simplity your answer Type integar fraction an m or an improper 507 pxal

Sketch the region enclosed by y 4x and y 5x Find the area of the region

26 find the general solution for the differential equ 7y2 3x option A y 2x 3 C B y ZX c c Y 2x c D y 7x c w praw XL2 EXP CS salt bod H A

6 Find the area under the curve y X over the interval 1 4

Evaluate the definite integral S 12C1nxj5 X dx

The pendulum in a grandfather clock makes one complete swing every 2 seconds The maximum angle that the pendulum makes with respect to its rest position is 6 We know from physical principles that the angle between the pendulum and its rest position changes in simple harmonic fashion Find an equation that describes the size of the angle as a function of time in seconds Take t 0 to be a time when the 0 pendulum is vertical and assume the pendulum is swinging toward its positive maximum angle Be sure to put parenthesis around your argument oc O

Evaluate the dernite integral 5x 4x 8 dx C 2

for a find the consumer Surplus if the demand functio particular beverage is given by D q 3000 and demand are in equilibam 3q 1 and if the supply q 6 o The consumer s surplus is 1

Evaluate the definite integral 8x 5 dx X Si 3 option A 13 51n3 B 16 51n3 C 51n 3 3 D 8 51n3

Calculate the iterated integral S 6xy x y dy dx

A contour map is shown for a function f on the square R 0 6 x 0 6 y 6 3 10 3 20 30 40 20 30 40 X a Use the Midpoint Rule with m n 2 to estimate the value of JJ f x y dA Round your answer to the nearest in b Estimate the average value of f Round your answer to one decimal place

Calculate the double integral 3xy 2 R X 1 J dA R x y 10 x 3 2 y 2 A

3 Power functions versus exponential functions Recall A power function is a function that can be represented in the form f x kx where k and p are constants and k is known as the coefficient 5 Let y f x x and g x 2 a Complete the table x y f x x y g x 2x 2 1 0 1 2 Is f x a power function Answer Is g x a power function Answer b Plot the points you found from part a 3 14 12 10 8 6 4 2 3 2 1 Note g x 2 is an exponential function with base 2 g x 14 12 10 8 6 4 2 X

Evaluate the integral using the following values x dx 320 x dx 16 dx 3 fra

Evaluate the integral using the following values x 8 x dx 960 x dx 24 x dx 24 dx 4 x 18 dx

iven F x f x dx 3 and Exercise a a Fx Therefore Step 1 To evaluate the integrals use the additive interval property of integrals which is given as follows Step 2 If f is integrable on the three closed intervals determined by a b and c then rb f x dx 2 evaluate the following f x dx Then To evaluate F x f x dx f x dx f f x dx 6 F x dx F x dx f x dx f x dx f x dx f x dx substitute f x dx 3 f x dx and f x dx 1 Xin

MY NOTES ASK YOUR TEACHER Sketch the region whose area is given by the definite integral 1 2x 3 dx 1 1 y 10 8 6 4 2 2 y 8 6 4 1 1 Need Help Read It 2 2 PRACTICE ANOTHER 3 Trapezoid Trapezoid 4 Use a geometric formula to evaluate the integral 3 5 4 6 X X 09 U 4 Trapezoid 6 5 4 3 Trapezoid 3 2 2 1 1 10 y y 8 6 8 6 4 2 2 1 1 X

Evaluate the integral using the following values L x dx x3 dx 1020 8 L 11 dx 8 8 x dx 30 dx 6

Use the substitution formula to evaluate the in tergral 6 3 St 3 j d 2

Evaluate the integral using the following values 8 x dx 1 020 x dx 30 dx 6 L 6x 1 11x 7 dx

Step 1 To evaluate Then Step 2 3 a 6 6 L 5f x dx use the following property of the definite integral 3 b To evaluate 5f x dx 5 Therefore 6 h 3 kf x dx k 3 6 6 S 5 3 5f x dx k 5f x dx 5 5 5f x dx substitute b f x dx 6 1 6 1 F 3 f x dx f x dx 6 f x dx in

Find the area of the shaded region 3 y 0 912 2 96 y 2 y x 9 x y 0 3 X

Sketch the region whose area is given by the definite integral 5 3 dx O O 5 y 4 y 3 2 18 16 14 12 10 8 6 2 2 3 X 4 3 5 Use a geometric formula to evaluate the integral O O 3 y 2 y 181 16 14 12 10 8 6 4 2 2 3 X 3 4 4 5 5 6 6

10 Use the integral test to determine whether the following series converge or diverge a n 5 n 1 b n 1 2n 1 n4 Hint Rewrite the integrand as 8 c n 2 Finally recall that lim arctan x x48 1 n ln n 2x 1 x T Hint Rewrite the function as In x Next use the fact that and arctan 1 J du 1 u arctan u and use u substitution to find the anti derivative The

Find the value s of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval Round you answer to four decimal places Enter your answers as a comma separated list f x 4 x 4 9 C 6 2500 X

Given that a b Rx dx f x dx 3 and 1 f x 1 F x f x dx 5 evaluate the following f x dx rx dx r x dx f x

Comparison of Companies Revenues The rate of change of the revenue of company A over the time interval U dollars week whereas the rate of change of the revenue of company B over the same period is g t dollars week The graphs of f and g are depicted in the accompanying figure Find an expression in terms of definite integrals involving f t and g t giving the additional revenue that company B will have over company A in the period 0 7 Ay S wk SE 1 Need Help y f t y g t T Select Select f t g t f t g t f t g t Read it f t T dt 71 wk Select dt

Use the Trapezoidal Rule and Simpson s Rule to approximate the value of the definite integral Compare your result with the exact value of the integral Express your answers correct to four decimal places 617x dx Trapezoidal Rule Simpson s Rule exact value n 4

LO O f x 3x and g x x x 1 5 5 a 10 20 10 20 30 S O e 5 30 e O 5 10 M e

3 S In 13x dx 1 3 s in 1 Do not round until the final answer Then round to four decimal places as nee In 13x dx

11 1 2 6 5 4 1 0 1 2 3 4 f t TT 2 TT 3m 2 5 2n 5m 2 31 Using the image above find a trigonometric function sin or cos for fo

1 Set up but do not evaluate the following triple integrals Use cylindrical or spherical coordinates if possible a E E x y z x y 3 z 5 x y 1 2r 2y dV where E is the region between paraboloids b a y sin zy dV where P is the prism drawn below 2 0 0 X Z 0 0 3 0 5 0 f

Evaluate the improper integral if it is convergent If it is divergent type DIVERGENT Lo x 8 00 x 9 DIVERGENT dx

Use the divergence theorem to determine the flux of F2 X Y Z X Y Z across the Sphere of radius P 2 centered at the origin

Na when x 4 per unit Du La Cu A firm has the marginal demand function D x At The demand function is D x 1800x x 25 where D x is the number of units sold at x dollars per unit Find the demand function given that D 12 000 did

The graph of y f x is shown to the right Graph y f x 4 Choose the graph of y f x 4 O A O C 10 Ay 40 Ay 10 X 10 110 57 OB 10 O D 10 10 Ay 10 10 Ay 2 10 10 10 Q 10 8 6 4 KWA IM ZET MIWA KWA f x Ay 8 6 4 2 NA 2 4 6 8 10 2 6 8 10 17

Evaluate the integral Round to the nearest tenth as needed 4 S7x11 2 7x In x dx OA 46 9 OB 64 9 O C 11 1 O D 6 7

a D x is the price in dollars per unit that consumers are willing to pay for x units of an item and S x is the price in dollars per unit that producers are willing to accept for x units Find a the equilibrium point b the consumer surplus at the equilibrium point and c the producer surplus at the equilibrium point OU La Cu At O D x 169 S x x x a What are the coordinates of the equilibrium point 169 13 Type an ordered pair b What is the consumer surplus at the equilibrium point Round to the nearest cent as needed

I use the definite integral to find the area between the x axis and the graph of FX over the indicated in tenal F x 2x x 0 2 option A 3 53 Nm um stm alm

For k 0 let Dk x y 0 x kr and a sin x y 2 and let Ak be the area of Dk Which of the following statement is truc a A 2 2 b A2 47 277 c A3 3 97 2 d A4 47 877 e None of the above statement is truc

1 Compute the following double integrals Be sure to show all steps a JJ 2xy 2 xy 3 dA where R is the rectangle R x y 1 x 2 5 y 2 b JL 2 c 2x y dA where T is the solid triangle with vertices 0 0 2 0 and 2 5 1 4r Au d4 where D is the disk r 1 6

Evaluate the dethite integral 2 5x 4x 8 dx 9

Find the length of the curve where r t 4ti cos t j sin t k and 0 t 34

7102 eigenvectors is 0 a has three distinct eigenvalues and one of its 0 0 A 0 1 0 c 0 0 Which one of the following can be anothe eigenvector of A 70372 b d POFTE

Evaluate the definite integral 1 xv X 48x 1 dx

O f x x 3x and g x x 3 y 10 8 6 4 2 20 10 8 6 20 10 2 6 8 10 8 6 4 2 10 20 10 10 20 y 20 10 J 4 246 8 10 10 2 4 6 8 10 20 O X y 10 O X 10 8 6 y 20 4 2 10 10 20 2 X 6 8 10 4 D

Find the following parameters for the figure shown a The expression or equation for this graph b Period of the graph c ao d an e bn and f Fourier expansion 8 SKA 6 6 8 12 12