Definite Integrals Questions and Answers

Find the total area between the x axis and the curve f x 2x 3 between x 5 and x 4 O O O 51 2 45 2 units units 136 3 units2 40 units 2

Use a Riemann sum with 4 rectangles of equal width to approximate the area between y x 2 and the x axis on the interval 1 1 Use the right hand endpoint of each subinterval O 3 5 units O9 units O 8 0625 units O 4 5 units

Write the definite integral x 3 dx as a limit of a Riemann sum and evaluate S O lim 7 n x k 1 7 32K 64x 2 1004 n n 3 O 64k lm 4 322 844 2 234 608 4 4 64 S COB 3 Olim 4 BAK 7 32 64 him lye dene n X K 1 3

Given f x dx O 7 S s f x 8 dx O 103 6 O23 5 15 find the following

Consider two functions f and g on 1 8 such that ff x dx 14 6 Sommer 9f x dx Simplify your answer a b f f x g x dx 1 C d 6 f f x g x dx e f Simplify your answer Simplify your answer f g x f x g x f x dx Simplify your answer 8 6g x dx 6g x dx Simplify your answer 6 6 5f x dx Simplify your answer 1 fg x dx 7 ff x dx 7 and g x dx 4 Evaluate the following integrals

Evaluate the integral sinx cos x dx using the fundamental theorem of calculus Discuss whether your result is consistent with the figure shown to the right 7x14 4 sinx cos x dx Is this value consistent with the given figure A The value is consistent with the figure because the area below the x axis appears to be equal to the area above the x axis B The value is consistent with the figure because the total area can be approximated using a rectangle of base x and height sin x 4 cos x 4 2 C The value is not consistent with the figure because the total area could be approximated using a rectangle of base x and height sin x 4 cos x 4 2 OD The value is not consistent with the figure because the figure is a graph of the base function f x instead of a graph of the area function A x 2 y sin x cos x

Inventory Control A company anticipates there will be a demand for 20 000 copies of a certain book during the next year It costs the company 0 50 to store a book for 1 year Each time it must print additional books it costs 200 to set up the equipment NOTE We assume that the demand is uniform Let 2 number of books printed during each printing run y number of printing runs Use this information to answer questions 9 19 below Question 9 Use the Inventory Control information above to answer this question 0 4 pts

the smaller area encloded by the curve y x and the circle x a 2 y 2 a 2

K If it snows tomorrow the probability is 0 7 that John will practice his trombone If it does not snow tomorrow there is only a 0 6 chance that John will practice Suppose the chance of snow tomorrow is 60 What is the probability that John will practice his trombone

Evaluate the line integral Vp dr for the following function op and oriented curve C a using a parametric description of C and evaluating the integral directly and b using the Fundamental Theorem for line integrals C r t sint cost for 0 st t p x y xy a Set up the integral used to evaluate the line integral using a parametric description of C Use increasing limits of integration dt Type exact answers

Determine whether the following vector field is conservative on R F 2y 2x 2y Is F 2y 2x 2y conservative on R O Yes O No

Solve each inequality log12 2x 1 log12 5x 16 O xx 5 O no solution 0 x s Question 5

4 Compute fF ds where F cos xy zy sin ry i x sin ryj and c t et cost et sint 0 t

O 6 5 O 2 2 0 1 1 15 3 02 12 4 3 1 6 5 3 0 1 2 3 4 5 6 1 2 Use the graph to estimate the slope of the line tangent to the given graph at 3 2 3 4 5 6

The region in the first quadrant enclosed by the graph of y x and the vertical line x 9 is rotated about the x axis What is the volume of the solid generated

A cable hangs between two poles of equal height and 40 feet apart At a point on the ground directly under the cable and x feet from the point on the ground halfway between the poles the height of the cable in feet is The cable weighs 12 pounds per linear foot Find the weight of the cable h x 10 0 5 2 5 HINT The limits for the problem should be 20 to 20 but the 20 would place a negative number under a square root Sc advantage of the symmetry use the limits 0 to 20 and double your answer

To find the length of the curve defined by y 5x5 8x from the point 3 1239 to the point 2 176 you would compute f x dx where a and f x b a

For the following exercises graph the equations and shade the area of the region between the curves Determine its area by integrating over the x axis 7 y x and y x 18x

For the following exercises determine the area of the region between the two curves by integrating over the y axis 5 x y and x 9 YA 4 2 0 2 4 2 x y 4 6 8 X 9 10X

0 5x The rate at which a substance grows is given by R x 185e where x is the time in days What is the total accumulated growth during the first 1 5 days K Set up the definite integral that determines the accumulated growth during the first 1 5 days 1 5 Growth S 1850 5x dx 0 Type exact answers in terms of e Evaluate the definite integral BELLEB Growth Do not round until the final answer Then round to one decimal place as needed

The length of the curve defined by the equation cy f x 16 from x a to x b is given by L f f x dx wher

Use the definite integral to find the area between the x axis and f x over the indicated interval Check first to see if the graph crosses the x axis in the given interval f x 4x 16 3 7 The area between the x axis and f x is Type an integer or a simplified fraction

Find the standard deviation o of the data 3 4 5 7 10 12 15 x 8 Variance 17 1 0 Round to the nearest tenth Standard Deviation Entor

The length of the curve defined by the equation xy 16 from x a to x b is given by L f f x dx where f x

Find the arc length of the graph of the function f x 4 from x 3 to x 5 arc length

Find the exact area between the graphs of f x e 1 and g x 3x 4 between x and x 2 O a 2 2 6 units b 0 0 units 9 1 2 c 0 0 units e d e units 1

You have answered 0 out of 2 parts correctly The following table gives the the approximate amount of emissions E of nitrogen oxides in millions of metric tons per year in the US Let t be the number of years since 1940 and E f t a Using left endpoints estimate the integral 5 5 f b Using right endpoints estimate the integral 6 5 0 ft Be sure that you know what the units of your answer are and what its meaning is f t dt 690 f t dt 815 t 1940 7 4 1950 1960 1970 1980 1990 9 3 13 3 18 1 21 3 19 6

A particle moves with a velocity of v t ft s along an s axis Find the displacement and the distance traveled by the particle during the given time interval Give exact answers v t t t 0 t 4 a Displacement 2 67ft b Distance 8ft include units include units

In this problem you will calculate the area between f x 3r and the z axis over the interval 0 2 using a limit of right endpoint Riemann sums Express the following quantities in terms of n the number of rectangles in the Riemann sum and k the index for the rectangles in the Riemann sum a We start by subdividing 0 2 into n equal width subintervals zo 21 2 22 2 1 Zn each of width Az Express the width of each subinterval Az in terms of the number of subintervals Az b Find the right endpoints 21 22 23 of the first second and third subintervals zo 1 1 2 2 z3 and express your answers in terms of n 246 n n n 1 72 73 21 HT c Find a general expression for the right endpoint z of the kth subinterval 1 2 where 1 k n Express your answer in terms of k and n 24 11 d Find f z in terms of k and n f zk k3 13 e Find f z Az in terms of k and n f z Az lim 11 00 48 Enter a comma separated list k3 Area lim 8 4 f Find the value of the right endpoint Riemann sum in terms of n f 2x Az 12 g Find the limit of the right endpoint Riemann sum f xx Az 12

Let f x be the function defined by f x 4x 2e 5 3 having g x 5 0 x f x g x lim 8 x 4 cos It is known that lim and let g x be a differentiable function g x

In general research shows that social media is more persuasive when it O a duplicates the advantages of one way communication Ob matches the format of print journalism Oc duplicates the advantages of face to face communication O d matches the format of TV journalism

Find the average value of f x on the interval 3 6

Ef is continuous and 1 F 1 f t dt 16 f t dt 14 find the following integral

4 Evaluate 1077 2 919 2 dt using the Fundamental Theorem of Calculus Part 2 1017 2 969 2 dt help numbers

e the given transformation to evaluate the integral JJ 4x 16y da where R is the parallelogram with vertices 3 9 3 9 4 8 n 2 2 2 10 x u v y v 3u

Evaluate the following integral given f x f x dx x if x 1 if r 1

Evaluate the indefinite integral 3 8 8z 5z 4z dz C

Evaluate f x dx where f x 8x X 9 sin x 0 x

Evaluate the indefinite integral 15 5 dr C

Evaluate the definite integral 20 5 dar

A company is designing propane tanks that are cylindrical with hemispherical ends Assume that the company wants tanks that will hold 1000 cubic feet of gas and that th ends are more expensive to make costing 5 per square foot while the cylindrical barrel between the ends costs 2 per square foot Use calculus to determine the minimum cost to construct such a tank

5 Determine g x given that g x 12x2 and g 2 3 30x 4 g 1

6 Determine f x given that f x 60x 60x f 1 1 and f 1 6

4 Determine P x given that P x 6e 4 10x and P 0 6

Evaluate the following line integral C x 6y2 ds C is the line r s x 6y ds Type an exact answer for 0 s 2

A thin wire represented by the smooth curve C with a density p units of mass per length has a mass M Sp ds mass of the wire C x y y 3x 0 x 4 with density p x y 3 5xy ds Find the The mass of the wire is about Round to one decimal place as needed

Evaluate the following line integral fxy ds C is the portion of the unit circle r s cos s sin s for 0 s st The value of the line integral is Simplify your answer

Calculate the volume of a hemisphere of radius 5 using an integral in spherical coordinates Feel free to search online for assistance But you have to evaluate the volume with an integral not just use the formula for the volume of a sphere

As part of an experiment a student collects a set of data and plots the data on excel obtaining the following graph 01 N O 0 1 2 1 1 3 Which of the expressions below is the equation that represents this graph O In p 2T 3 O 2 In p 3 Oln p 2 7 6 T 2 ln p 6

Estimate the area under the graph of f x x 3x from x 5 to x 9 using 4 approximating rectangles and left endpoints Approximate Area