Definite Integrals Questions and Answers

Which of the following functions are solutions of the differential equation y y sin x Select all that apply O y cos x Oy x sinx 5x cos x O y sin x Oy x sin x Dy exco COS X

Find the average value of the function f x a 1 x on the interval 0 1 0 2 2 1 ONE 2 2 1 0 1 2 2

Find the length of the curve with parametric equations z 2t y 3t where t is from 0 to 1 O 2 2 1 04 2 1 08 2 4 04 2 2

Set up a definite integral for the length of the curve with parametric equations 2t y 3t where t is from 0 to 1 a You must show your work to receive full credit

Set up a definite integral for the volume of the solid generated by revolving the plane region bounded by y 1 y 0 x 0 x 1 about the y axis You must show your work to receive full credit

Consider the surface defined by the collection of points x y z R such that 3xz5 xy z 4 Show that z f x y near the point 1 0 1 After this compute the gradient vector V 1 0

Solve the differential equation dy dx 1 3 2n C 6x y for y 0 ANSWERS SCALCET9

Triangle ABC shown below has mZA 147 b 6 and c 16 Find the area of the triangle Round your answer to the nearest tenth and do not include units in your answer B a 16 147 A 6 C

Consider the following curve 2 sin 20 S Set up an integral that can be used to find the shaded region A Find the area of the shaded region de

Find the area of the shaded region r 1 cos 0 Q r 3 cos 0

Graph the curve r 6 sin 40 6 4 2 Need Help y 6 4 2 2 4 6 Find the area that it encloses Read It 2 4 Watch It 6 X 6 4 2 y 6 4 2 2 2 4 6 1 6 4 2 4 4 2 21 y 2 4 6 Xx 4 2 2

Sketch the curve with the given polar equation by first sketching the graph of r as a function of 8 in Cartesian coordinates 2 4 sin 28 y 3 2 8 X 3 3 2 2 2 3 O Need Help Read It Watch It 2 y 2 2 2 WA X 1 5 1 0 0 5 y 1 5 1 0 0 5 0 5 1 0 1 5 0 5 1 0 1 5 X 1 y 2 1 1

10 Find the exact area of the surface z x y in the 1st octant that lies between the cylinders x y 2 a x y 3 0 0 1 5 0 5 1 0 1 0 0 5 1 5 0 0 2

2z Use the Divergence theorem to find the outward flux of F In x y i tz tan zx y k across the boundary of the region D which is the thick walled cylinder 1 x y 2 1 z 2

f x x 27x 61 Identify all the relative minimum points Select the correct choice below and if necessary fill in the answer box to complete your choice OA The relative minimum point s is are Simplify your answer Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed OB There are no relative minimum points Identify all the relative maximum points Select the correct choice below and if necessary fill in the answer box to complete your choice OA The relative maximum point s is are Simplify your answer Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed OB There are no relative maximum points Graph the function Choose the correct graph below OA OB SELLE Q Q O C Q OD

Find the antiderivative F of f that satisfies the given condition Check your answer by comparing the graphs of f and F f x 2ex 8x F 0 5 F x I

In APQR sin P sin Q 3 and p 15 Find the side lengt Select the correct answer below 09 9 1 O q 75 09 6 O q 31 09 15 O q 30

Use a parametrization to express the area of the surface as a double integral Then evaluate the integral The portion of the cylinder x y 4 between the planes z 4 and z 7 Let uz and v 0 and use cylindrical coordinates to parameterize the surface Set up the double integral to find the surface area JO du dv 00 Type exact answers

Produce graphs of f that reveal all the important aspects of the curve Then use calculus to find the following Enter your answers using interval notation Round your answers to two decimal pla f x 8 sin x x 5 x 3 Find the interval s of increase Find the interval s of decrease Find the inflection points of the function smaller x value larger x value x y x y Find the interval s where the function is concave up Find the interval s where the function is concave down

Find f Use C for the constant of the first antiderivative D for the constant of the second antiderivative and F for the constant of the third antiderivative f t 24 cos t f t X Remember to use capital C D and F

10pts Section 16 2 Evaluate J Laye x y 2 Include a sketch of the region of integration in your solution xy dA where R is the region bounded by the lines y x y 2x and

Find any absolute or local maximum and minimum values of f x 8 2x if x 6 Select the correct answer O 4 is a local maximum 4 is an absolute minimum 6 is an absolute minimum O 4 is an absolute maximum 6 is an absolute maximum 6 is a local minimum

1 point a Find the general solution to y 3y 0 Give your answer as y In your answer use c and c to denote arbitrary constants and x the independent variable Enter c as c1 and c as c2 help equations b Find the particular solution that satisfies y 0 1 and y 0 1 help equations

Vrea sin gr dr ria Vore sin gr dr Va rla a q 1 aq a q r 2a r cos gr a q a q r a r sin qr 2V a q a q 1

Which expression below can be obtained from 22sin a by using a power reducing formula Select the correct answer below 11cos 40 11cos 2a cos 40 11cos 20 cos 4c 11eos 2g 11cos 4a 33

6 Calculate 5x dx given the following 6 4 5 x dx 4 61 3 15x dx 0 2 6 2 Sx dx 5 91 3 5 9 xdx 1 2 4 4 Type an integer or a simplified fraction

x 2 2 9 y 2 1 2 x 3 y 0 y 1 3 sqrt x 2 441 east one of the answers above is NOT correct dy X dx 9y point Solve the differential equation a Find an implicit solution and put your answer in the following form x 2 2 9y 2 2 constant help formulas Find the equation of the solution through the point x y 3 1 x 3y help equations x 9y 2 2 x 3y 0 y 2 441 Find the equation of the solution through the point x y 0 7 Your answer should be of the form y f x y 1 3sqrt x 2 441 help equations correct correc incorre

Simplify the lefthandside so that LHS sin a sin a 1 cos a 1 cos a RHS 2 tan a

1 point Solve y sin x if y 0 0 and y 0 3 y x help formulas

1 Cars arrive at a one bay car wash facility where the interarrival time is exponential with mean of 10 minutes Arriving cars line up in a single line It takes between 10 and 15 minutes uniformly distributed to wash a car Assume first arrival occurs at time 0 T 0 and system is empty initially Simulate the system for first 7 customers and calculate the following performance measures a What is the average queue length 20p b What is server utilization 20p c What is the average time that a customer waits in the queue 20p X Ln R Table 1 Random numbers x 9 a 6 a R Random Numbers for Interarrival times Random Numbers for Service times 0 3397 0 1351 0 2677 0 8762 0 9912 0 8176 0 2236 0 8422 0 8820 0 8471 0 4249 0 9113 0 9387 0 3456 2 Using multiplicative congruential method generate 10 random numbers for following input parameters 20p

You are given that cos A your answer as a fraction 13 with A in Quadrant III and sin B 13 with B in Quadrant I Find sin A B Give

The equation p 5000cos t 54000 describes the number of deer in a forest where t is the number of years af 1972 What was the population in the year 1991 to the nearest whole number NOTE The angle is in radians Provide your answer below

Considering only the values of for which sec B 1 cos 1 cos B is defined which of the following expressions is equivalent to see 1 cos B 1 cos Select the correct answer below 1 sec tan cos B sec sin tan

onsider the given function and the given interval f x x 6 5 8 a Find the average value fave of fon the given interval fave b Find c such that fave f c C C c Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f f x f x 4 3 smaller value larger value 2 2

The graph shows the function y h t 1 5 4 3 2 1 X 1 2 3 4 5 a I We define a new function r x h t dt 0 2 Complete the table of values for the function r 1 5 r x te

5 Evaluate the definite integral e4 dx xvlnx

4 Write as Partial Fractions x 6 x 7

Use the substitution t tan 2 6 sin 2x 2 cos x dx to transform the integrand into a rational function of t and then evaluate the integral

Find the potential function f for the field F 6x i 7y j 9z k 2012 TRONISAT A general expression for the infinitely many potential functions is f x y z

A car rental agency rents 440 cars per day at a rate of 40 per day For each 1 increase in rate 10 fewer cars are rented At what rat should the cars be rented to produce the maximum income What is the maximum income The rental agency will earn a maximum income of when it charges per day

Find a potential function for F 3 4x 2 8x F i y y j x y y 0 A potential function for F is f x y z

Step 4 Since the square root and the square cancel each other we have 3 L e e d dt Submit 11 3 1 Skin you cannot come back

Evaluate f x y ds x y ds where C is the straight line segment x 4t y 8 4t z 0 from 0 8 0 to 8 0 0 f x y ds

Tutorial Exercise Find the exact length of the curve x et e t y 5 2t Ost 3 Click here to begin

Find the line integral of f x y ye along the curve r t 3t i 4tj 2sts 1 The integral of f is Type an exact answer

Find the line integrals of F 4yi xj 2zk from 0 0 0 to 1 1 1 over each of the following paths a The straight line path C r t ti tj tk 0st 1 b The curved path C r t ti t j t k Osts1 c The path C3UC4 consisting of the line segment from 0 0 0 to 1 1 0 followed by the segment from 1 1 0 to 1 1 1 0 0 0 C 1 1 1 CA 1 1 0 a The line integral of F over the straight line path C is Type an integer or a simplified fraction

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y axis y y 0 x 2 V Sketch the region and a typical shell y 10 8 6 4 X 3 2 4 X 2 10 2 y 8 6 2 2 4 X

Evaluate f x y x y ds where C is the straight line segment x 2t y 8 2t z 0 from 0 8 0 to 8 0 0 C x y ds C Type an exact answer

Integrate f x y z The integral of f is x y z over the path r t 2t i 3t j tk 0 astsb 2 x y z

Evaluate fc xy x xy x z ds along the curve r t 2ti tj 5 2t k 0 st 1 c xy c xy x z ds Simplify your answer ww