Definite Integrals Questions and Answers

For each of the following determine if the series converges Be sure that you can rigorously demonstrate your answer A 4n 7n 1 n 1 This sum A converges O B diverges B 3 7 15 n 1 This sum O A converges OB diverges c n 7 72 0 This sum OA converges OB diverges D n n8n 16 n 7 This sum A converges B diverges

Find the value of 8 1 9n 2 7 n 2 Enter if series is convergent D if series is divergent Determine whether rx L 2 is a convergent series dx 9x 2 7

Let R be the region bounded by the curves y 1 x 2 y 0 x 1 and x 5 What is the volume of the solid generated by rotating R about the x axis

Let f x y x y and c t t 2t a Calculate Vf c t 3x y 21 x 61 b Use the Chain Rule for Paths to evaluate f c t at t 1 f c 1 18 Answers Attempt 5 of 5 Answer

Draw a picture and set up the integral using the method specified DO NOT INTEGRATE Draw a picture and set up the integral to find the volume of the solid generated by revolving the region bounded by the graph of x y 16 x 0 about the x axis Shell Method To enter your answer click on the three vertical dots on the right of the task bar then click on the symbol This will bring up the equation editor Use the symbols to write the mathematical sentence Drawing and work will be graded separately Your Answer

Evaluate the following Find the volume of the solid formed by revolving the region bounded by the graphs of y 1 and y 2x about the line y 1 To enter your answer click on the three vertical dots on the right of the task bar then click on the symbol This will bring up the equation editor Use the symbols to write the mathematical sentence Work will be graded separately Your Answer 74T 15

Find the critical point of the function f x y x y 4xy 42x C Use the Second Derivative Test to determine whether the point is O A a local maximum OB a local minimum OC test fails O D a saddle point

Find the equation of the circle shown in the figure LAHA 3 2 1 y 3 1 1 2 X

Rewrite with rational exponents 5 3 xy z 5 xy Sim

At what point does the line normal to the graph of y 3x 3 4x at 1 4 intersect the parabola a second time 0 Hint The normal line is perpendicular to the tangent line If two lines are perpendicular their slopes are negative reciprocals i e if the slope of the first line is m then the slope of the second line is 1 m

A golf ball is hit with a driver and neglecting almost all physical forces except gravity follows the path y 300 x I 600 2 The ball is on the ground at x 0 and at x 300 We will assume that the ball did not roll when it hit the ground We would normally say that the drive was 300 yards long which is the distance along the ground that the ball traveled The actual distance that the ball travels is given by the arc length of the parabola The arclength is given by the integral 300 1 y dx 1 x 300 1 2 dx Problems 1 Evaluate the integral above using an electronic device How far did the ball travel 2 The graph of a function f x is given below It consists of three line segments for 0 x 8 Each mark along the x axis and y axis is 1 unit a Calculate the total length of f x for 0 x 8 b What is f x for 0 x 3 c What is f x for 3 x 4 d What is f x for 4 x 8 f x e The graph of y 1 f x is given below Calculate the y coordinates of the three line segments using your answers from parts b d f Find the areas of the three rectangles in the second figure below using your answers from part e 1 f x

nd the intervals on which the function is continuous 17 y 2 X 7 2x A continuous everywhere C discontinuous only when x 9 B discontinuous only when x D discontinuous only when x

Submit Answer DETAILS SCALCET9 6 1 004 3 1 Points Set up an integral for the area of the shaded region Evaluate the integral to find the area of the shaded region 10 5 5 x 4y y x y 6 y y 5 5 5

The electric charge q in C passing a given point in a circuit is given by q t sec 0 21 t 0 60 s idq dt A Thon round to two decimal places as needed 1 where t is the time in s Find the current i in A for

Integrate the given function 5 3 dy y y Simo dy Then round to two decimal places as n

Find the area bounded by the following curves y 9 x y 0 x 1 x 2 The area is plify your answer

Use the shell method to find the volume of the solid generated by revolving the shaded region about the y axis The volume is Type an exact answer using as needed 6 2 y 21x 1 18

ind the coordinates of the centroid of the region bounded by y 49x and 1 The region is covered by a thin flat plate What are the coordinates of the centroid Type an ordered pair

33 42 Composite functions and notation Let f x x 4 g x x and F x 1 x 3 Simplify or evaluate the following expressions 33 g 1 z 35 F g y 34 F y 36 f g w f 2 h f 2

Find the first quadrant area under the curve of y The area is units Round to three decimal places as needed 3 4 In x X from x 1 to x 5

Integrate the given function dx 7 2x dx 17 2x 0 7 2x

Approximate the value of the given integral by completing parts a and b below a Use Simpson s rule using n 2 b Check by direct integration 2 S 6 x dx 0 a Use Simpson s rule using n 2 2 Sco 31

Find the equation of the curve that passes through 3 4 if its slope is given by the following equation dy dx 3x 7 y x 0

Evaluate 16 16 1 4t 1 2 1 4t 1 1 2 t dt dt Simplify your answer

Find the length of the following curve 312 y 2x from x 0 to x 4 The length of the curve is Type an exact answer using radicals as needed

Find the length of the following curve 3 2 y 2 X y from y 7 to y 9 3 3 N W 2 1 2 y The length of the curve x 3 Tyne an exact answer using radiolo from y 7 to y 9 is

The graph of the equation x2 3 y2 3 102 3 is one of a family of curves called astroids because of their starlike appearance as shown in the accompanying figure Find the length of this particular astroid by finding the length of half the first quadrant portion y 102 3 2 3 3 2 3 2 5 2 2 x 10 and multiplying by 8 The length of the given astroid is Simplify your answer Type an exact answer using radicals as needed 10 10 10 10

Find the length of the following curve If you have a grapher you may want to graph the curve to see what it looks like 32 y 27 9x 6 from x 3 to x 9 The length of the curve is Type an exact answer using radicals as needed

4 3 U9 Carbon dioxide is produced by combustion of biomass and fossilized organic matter Define combustion State the products of a combustion reaction State sources of fuel for a combustion reaction 4 3 A2 Analysis of data from air monitoring stations to explain annual fluctuations Sketch a graph of the annual fluctuation in atmospheric carbon dioxide concentration Paste here Explain the annual fluctuation in atmospheric carbon dioxide concentration in the northern hemisphere

raph a line that is perpendicular to the given fine Determine the slope of the given line and the one you graphed in simplest to lick and drag on the graph to draw a line Click and drag to plot a perpendicular line The line will change colors when a parallel or perpendicular line is drawn accurately Answer Attempt 1 out of 2 TR 719 Ter 3

The acceleration m of an object along a straight line is given in the following graph sec2 Acceleration 10 5 1 0 1 Time OA 53 5 meters OB 28 5 meters OC 27 0 meters OD 19 5 meters OE 33 0 meters 2 Assuming that the object has an initial velocity of 5m how far does the particle move during the time 0 t 3 sec E

17 For the function g whose graph is given arrange the follow ing numbers in increasing order and explain your reasoning 0 g 2 g 0 g 2 g 4 0 y g x 2 3 4 www X

Using a left endpoint Riemann sum approximation with four equal subintervals the approximate value of fo g x dx is A 8 B 8 C 16 OD 16 01 2 3 4 5 6 7 8 g x 1 2 5 3 1 5 1 25 0 1 5 4 5 X E 13 5

a Find the slope of the tangent line to the parabola y x 3x at the point 1 2 i using Definition 1 ii using Equation 2 b Find an equation of the tangent line in part a c Graph the parabola and the tangent lin As a check on your work zoom in toward the point 1 2 until the parabola and the tangent line are indistinguishable

The rectangle shown below has vertices of 0 0 0 2 1 and 0 1 y cos x What percent of the rectangle is shaded OA 63 7 B 36 3 OC 31 4 OD 50 OE 43 2

If the definite integral e da is first approximated by using two inscribed f rectangles of equal width and then by using the trapezoidal rule with n 2 the difference between the two approximations is OA 26 80 OB 53 60 OC 27 80 OD 12 78 O E 30 51

For which value of x does the function f x x 2 x 3 have a relative maximum A Both 2 and 3 B 3 C 2 OD 5 2 OE O

3 Solve the following Initial Value Problems IVP and find the interval of validity for the solution of each problem a y c y dy b e sec y 1 1 y 0 0 dt e 3x 4x 4 2y 4 dy dt 3 xy 1 x dy d 6y x y 1 dx y 1 3 y 0 1 1 25 y sint y 0 1 5 5 5 5 5

13 16 Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a 13 f x 3x x 2 5 a 14 g t t 5t 2t 1 a 2

Consider the position function s t 4 9t 291 17 Complete the following table with the appropriate average velocities and then make a conjecture about the value of the instantaneous velocity t 4 Complete the following table Type exact answers Type integers or decimals Time Interval Average Velocity 4 5 4 4 1 4 4 01 4 4 001 4 4 0001 15 1 10 69 10 249 10 2049 10 2005

Consider the following graph of y f x 40T 2 20 0 20 40 60 2 4 O A I only OB II only O C III only OD II and III only E I II and III 6 8 10 Which of the following could be the equation for f x I A Ba with 0 a 1 B 0 and A 0 I A Ba 2 with a 1 B 0 and A 0 III A a B with 0 a 1 B 0 and A 0

The average value of the function g x 3cos on the closed interval T is OA 7 593 OB 4 347 O C 1 325 OD 18 068 OE 30 980

If f 2 2 5 and f 2 2 5 then f 2 5 is approximately OA 2 5 OB 2 OC 2 5 OD 1 25 OE 2

Jo 1 4 O A In 1 4 O B jarctan 24 O C j arcsin 22 O D In 1 x4 O E arctan z

1 The shaded regions R and R shown below are enclosed by the graphs of f x x and g x 2 R R I A Find the x and y coordinates of the three points of intersection of the graphs of f and g B Estimate the area of R to three reasonable places past the decimal C Find the EXACT area of R No calculator approximations

2 The region R is a rectangle with vertices P a Ina Q a 0 S 3 0 and T 3 Ina where 1 a 3 201 1 54 05 1 0 P T 10 15 20 25 30 0 S A Write an expression that gives the area of the rectangle as a function of a B The area of the rectangle is maximized for some c between 1 and 3 Write the expression you would need to solve in order to find c You don t need to find c C What is the rate of change of the area when a 3 if a is decreasing at a rate of da 0 5 units sec You don t need to simplify of course Use appropriate units dt

O A e 3 C Fe O B e C OC 255 22 C O D e 3 C O E Fe C

Suppose s t is the position of an object moving along a line at time t 0 What is the average velocity between the times t a and t b Which of the following is the correct formula for average velocity OA Vav OB Vav OC Vav wer Vav s a s b a b s b s a b a s a s b b a s b s a b a

For what values does the function fail to exist f x A x 1 4 and x 0 C x 4 x 0 and x 4 X x 16x B x 1 4 x 0 and x 4 D x 1 4 x 0 and x 4

For this question refer to the following graph and information An ant is crawling on a straight wire The VELOCITY v t of the ant at time 0 t 8 is given in the graph at right Note The graph on 0 t 2 is a semicircle half circle A 4 2 B 5 7 C 8 1 2 D 5 1 2 E 37 5 3 What is the net distance traveled by the ant over the time interval 0 t 8 Click here for long description