Definite Integrals Questions and Answers

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

Question 1 The bounded region shown for each problem repre on the given cross sections Set up the integral s fi 3 y x and y 2x

LINE Google Slides estion 5 ut do not solve MGmail find the area of the region bounded by the fol Y v r 6 v 2 Spotify

5 Consider the graph below Assume that h x cx Determine the value of c so that the area on the left is the same as the area on the right for all positive x values You should simplify c to a rational number Jychenr 7 2x2 y x

A stone is thrown straight up from the edge of a roof 975 feet above the ground at a speed of feet per second A Remembering that the acceleration due to gravity is 32 feet per second squared how high is the stone 3 seconds later B At what time does the stone hit the ground C What is the velocity of the stone when it hits the ground

Given f x 25 sin 5x and f 0 5 and f 0 3 Find 6

6 Graph of f The derivative of a function fis defined by f x X g x for 4 x 0 5e 2 3 3 for 0 x 4 The graph of the continuous function f shown in the figure above has x intercepts at x 2 and 2 3 ln The graph of g on 4 x 0 is a semicircle and f 0 5 a For 4 x 4 find all values of x at which the graph of fhas a point of inflection Justify your answer b Find 4 and 4 c For 4 x 4 find the value of x at which fhas an absolute maximum Justify your answer

The graph of the function fabove consists of three line segments BI U 22 1 2 2 4 3 1 2 Let g be the function given by g x f t dt For each of g 1 g 1 and g 1 find the value or state that it does not exist x x 5 C E Graph of f 2 0 3 2

1 2 4 3 1 2 0 Graph of f 2 0 3 2 The graph of the function fabove consists of three line segments For the function g defined in part a find the x coordinate of each point of inflection of the graph of g on the open interval 4 x 3 Explain your reasoning

Find the derivative of the function g t tan cos 6t 6 sin 6t sec cos 6t 6 sin 6t sec cos 6t sin 6t sec cos 6t sin 6t sec cos 6t

f x Ja 0 2 0 Below evaluate f a at the points indicated in the graph help numbers Enter DNE if the function is not differentiable at the given point Identify the points as being an absolute maximum or minimum a relative maximum or minimum or none of the above Select all that apply A X A 0 5 0 25 f 0 5 At A there is a an A Absolute maximum B Absolute minimum C Relative maximum D Relative minimum E None of the above B 0 0 f 0 At B there is a an A Absolute maximum B Absolute minimum C Relative maximum D Relative minimum E None of the above C 0 4 0 01024 f 0 4 At C there is a an OA Absolute maximum B Absolute minimum C Relative maximum D Relative minimum E None of the above

In each part a function f x and interval a b are given Check if Rolle s Theorem can be applied to f on a b a f x 11 on 1 1 f f f 1 Rolle s Theorem continuous on 1 1 differentiable on 1 1 and f 1 b f x 112 on 1 1 f f f 1 continuous on 1 1 differentiable on 1 1 Rolle s Theorem apply in this situation and f 1 c f x 11 x on 1 1 f f 1 apply in this situation continuous on 1 1 differentiable on 1 1 and f 1 The two values are The two values are The two values are

Consider the following indefinite integral The integrand has partial fractions decomposition where a b C d 5x 8x 27x 45 x 9x Now integrate term by term to evaluate the integral a b X x cx d x 9 dx

Find f s if we know that s 12s s 1 2 3 and C

Graph of f g x for 4 0 5e 3 3 for 0 1 4 The graph of the continuous function f shown in the figure above has x intercepts at x 2 and a 3 ln The graph of g on 4 sxs0 is a semicircle and f 0 5 The derivative of a function fis defined by f x a For 4 x 4 find all values of x at which the graph of fhas a point of inflection Justify your answer B U x CEE

y vertical or oblique s in the graph of ate the domain of 12 8 1 D U Ellb Tiz IR 1 1 IT 1 42 U 16 11 10 11 10 4 11 1 1 LE 4 I 1 I T X 8 12 16 20 Identify any horizontal or oblique asymptotes Select the correct choice below and if necessary fill in the answer box to complete your choice O A The function has a horizontal asymptote whose equation is Type an equatic B The function has an

y vertical or oblique is in the graph of ate the domain of 2 8 JO T n 14 6 18 I D 21 n 10 in 16 1 16 4 h I 1 I T H 14 I I I 1 112 U T 12 16 20 4 GUIT y 5 Type an equatic B The function has an oblique asymptote whose equation is Type an equation OC The function has neither a horizontal nor an oblique asymptote The domain of f is Type your

hamed Salad Attempt 1 Question 5 0 833 points Listen Determine the area bounded by the given curves y sin TT y x 6x O 54 12 none of these 16 12

Find the open intervals where the function is concave upward any inflection points

nd the indefinite integral using integration by parts with the given choices of u and dv Use C for the constant of integration x x4 In x dx u In x dv x4 dx

Use a numerical integration routine to 2 1 2 6 3 7x dx

The mean value theorem states that if F x is a differentiable function on the interval a b then there exists some number c between a and b such that the following formula is true Use this to explain why the mean value theorem implies that if a car averages 58 miles per hour in some 25 minute interval then the car s instantaneous velocity is 58 miles per hour at least once in that interval F c F b F a b a Choose the correct answer below OA The car s speed of 58 miles per hour is defined as the average rate of change of the car at point c or f c Since car drives continuously over a 25 minute interval or 0 25 the function is continuous Thus the mean value theorem can be applied OB The car s speed of 58 miles per hour is defined as the average rate of change of the car at point c making the function continuous Since the function is continuous it has a derivative at every point Thus the mean value theorem can be applied C The car s time of 25 miles per hour is defined as the average rate of change of the car at point c making the function continuous Since the function is continuous it has a derivative at every point Thus the mean value theorem can be applied OD The car s time of 25 miles per hour is defined as the average rate of change of the car at point c or f c Since the car drives continuously at an average speed of 58 miles per hour the function is continuous Thus the mean value theorem can be applied

Mohamed Salad Attempt 1 Question 3 0 5 points Listen Evaluate 4x x 3 4 dx O 3 5 C O 3 5 C x 3 5 C O x 3 5 C 2 5

Answer the following questions about F x 3x 180 A Calculate the change in F x from x 8 to x 16 B Graph F x and use geometric formulas to calculate the area between the graph of F x and the x axis from x 8 to x 16 C Verify that your answers from A and B are equal as guaranteed by the fundamental theorem of calculus A Calculate the change in F x from x 8 to x 16 The change is Simplify your answer

For the function f x 3x 4x 11x 7 find t x Then find t To find f x first find f x f x Find f x f x Find f 0 Select the correct choice below and if necessary fill in OA 1 0 OB f 0 is undefined Simplify your answer

For a standard normal distribution given P Z c 0 1898 Find c

2 Calculate the area of the region bounded by the curves y x y x 2 and the lines y 0 and y 2

39 Given A 24 25 B 24 C 2 44 7 D 24 25 sine 7 25 which is in quadrant I find tane

24 Evaluate the expression cos Tan A 5 B 12 13 C 1 2 13 D 5 13

OC Some people are being less than honest because P x 2 768 is less than 5 O D People are being honest because the probability of P x2768 is less than 5

Calculate the indicated Riemann sum S3 for the function f x x 6x 27 Partition 0 6 into three subintervals of equal length and let c 0 8 C 2 8 and S3 C3 c3 4 5

5 Let f x c 1 x a For what value of c is f a probability density function b For that value of c find P 1 X 1

If sin 0 cos 0 and is in the 1st quadrant find the exact value for cos 0 8

sin COS tan 1 0 2 3 ans 1 The point P is on the unit circle If the y coordinate of Pis Give an exact answer rather than a decimal approximation 1 and P is in quadrant III find x 2

0 ans 1 The angle between 0 and 360 that is coterminal with the 139 angle is degrees

er 7 Group Worki The following account balances come from t Accounts receivable Allowance for doubtful accounts During the accounting period Ourso record receivable

Given fox dx 18 Calculate 3x dx 54 Jo x dx 72

The indefinite integral can be found in more than one way First use the substitut integral Then find it without using substitution Check that your answers are equ 9x x 8 dx D X 8 Find the indefinite integral 1 9x8 x 8 dx x 8 c Find the indefinite integral without using substitution 1 fox x 8 dx x1 2 Are the two answers equivalent If not how do they differ x18 8x9 C On The two answers are the same

The marginal price for a weekly dem Find the price demand equation if th

Find the antiderivativ ds dt 11t 4t 7

7 a Find the area of the region bounded by the curves y x and y x b Find the volume of the solid formed by revolving the area from a about the line y 1

b Hence find the exact values of the definite integrals of each of the following 1 i S 3 2x 3 dx 0

bau se episode soutenisulodd some with a 3 dianomis Caloo 2 Calculate the area of the region bounded by the curves y x y x 2 and the lines y 0 and y 2 Tal dot D

Given a b c F x f x dx 8 and f x dx 7 evaluate the following F x f x dx f x dx F x 3 f x dx d 5f x dx

In 2 3 without solving state whether the triangle has one two or no solutions Justify your answer 4 2 AFGH GH 22 FH 39 FG 19 3 ADEF EF 16 DF 5 mzB 42

Write the partial fraction decomp 2 X x 10x 9

21 26 Set up an integral for the area of the surface obtained by rotating the given curve about the specified axis Then evaluate your integral numerically correct to four decimal places 21 y ex 1 x 1 x axis

17 20 The given curve is rotated about the y axis Find the are of the resulting surface 17 y x 2 0 x 12 18 x2 3 y2 3 1 0 y 1 19 x a y 0 y a 2

Evaluate Assume u 0 when In u appears fin 9x dx O O O O 7x 3 4 C 56 7x 3 6 C 7x 3 6 C 7x 3 4 C

x 1 0 Let f be the function defined by f x For f x cos 3t5 dt Find an expression