Definite Integrals Questions and Answers

Find a definite integral for which 1 4 4 k 4 4 4 is a right rectangle approximation 5 O A x dx B x 4x dx c x 4x dx OD 21 6 5 O E x dx

1 Which of the following will give the smallest value for da for a given number of intervals A A right hand Riemann sum approximation B A trapezoid approximation OC The actual value of the definite integral D A left hand Riemann sum approximation E A midpoint Riemann sum approximation

The region Ris bounded by the x axis x 2 and y x Which of these expressions represents the volume of the solid formed by revolving R about the line x 2 OA ST ydy B 2 x dy c f 2 x dy OD ST 2 y dy OE ST 2 y dy

If f 3x x 1dx udu then So S a A a 2 b 4 OB a 8 b 64 OC a 12 b 48 OD a 7 b 63 O E None of these

4 If f 3x x 1dx udu then 2 a O O A a 2 b 4 B a 8 b 64 C a 12 b 48 OD a 7 b 63 OE None of these

Graph the function f x over the given interval Partition the interval into four subintervals of equal length Then add to your sketch the rectangles associated with the Riemann sum f ck Axk given that ck is k 1 the a left endpoint b right endpoint and c midpoint of the kth subinterval Make a separate sketch for each set of rectangles f x sinx 4 b Sketch the rectangles associated with the Riemann sum f ck Axk given that ck is the right endpoint of the kth subinterval Choose the correct graph below k 1 O A O E AL OB O F Ay N IN G O C O G Ay Ay Q Q Q O D OH Ay Q

4 k 1 Graph the function f x over the given interval Partition the interval into four subintervals of equal length Then add to your sketch the rectangles associated with the Riemann sum 1 ck Axx given that c is the a left endpoint b right endpoint and c midpoint of the kth subinterval Make a separate sketch for each set of rectangles f x sinx OA E OE c Sketch the rectangles associated with the Riemann sum f ck Axk given that ck is the midpoint of the kth subinterval Choose the correct graph below B 4 O F k 1 E Ay S O C O G Ay n 14 Ay S O D OH DUN Ay 0004 Q

4 Graph the function f x over the given interval Partition the interval into four subintervals of equal length Then add to your sketch the rectangles associated with the Riemann sum f c Axk given that c is k 1 the a left endpoint b right endpoint and c midpoint of the kth subinterval Make a separate sketch for each set of rectangles f x sinx n a Sketch the rectangles associated with the Riemann sum f ck AxKk given that ck is the left endpoint of the kth subinterval Choose the correct graph below k 1 O A 1 O E Q O B 4 O F Ay Q O C O G AY Ay Q O D OH Ay DON

Find the volume of the following solids The base of a solid is the region between the curve y 8 sin x and the interval 0 on the x axis The cross sections perpendicular to the x axis are a equilateral triangles with bases running from the x axis to the curve as shown in the figure b squares with bases running from the x axis to the curve y 8 sin x

Evaluate the limit r t h r t h lim h 0 r t for r t t sint 7 2 41

The area of the region enclosed by the curves y given by which of the following expressions O A S O C OE 0 28166 1 67738 2 26296 2 26296 O B S 1 67738 2 x 3x dx 1 67738 0 28166 x 3x 2 dx 2 x 3x dx 1 67738 x 3x 2 dx 2 26296 028108 2 2 3 32 0 28166 9 85917 2 26296 O D 1 67738 2 26296 228296 0 28166 2 x x 3x 2 dx 0 28166 1 67738 3x dx dx x 3x 2 dx 2 x 3x dx 3 and y 1

Evaluate the integral 5si 9s j 13k

Write an equation of the plane with normal vector n 1 2 7 passing through the point 6 9 9 in scalar form 204

Find the area of the region bounded by the curves y x y 4 x y 2 and y 1 Hint You ll definitely have to sketch this one on paper first You get O A 27 OB 14 O C 2 O D 22 O E 3 4

Find the area of the region in quadrant I bounded by the curves y x 2 y x 2 y 1 and y 3 1 2 You get O A 3 O B 2 3 O c 2 3 O D 2 3 32 O E 3 3 8 coco

Find the area between the curves y f x x 3x and y g x x 4x x on the interval 0 x 2 O A 200 OB B 83 O C 4 O D O E

Find the area of the region bounded by the curves 3 y sin x y csc x x 4 and x You get O A 2 2 OB 2 2 O c 2 2 OD 2 2 OE 2 2

1 The popu present population A Model the population P P 1 of the bacteria population with a differential equation B Solve the differential equation that models the population P P t of the bacteria population t C Suppose the population at time 0 hours is 1000 Derive an equation for the population P as an explicit function of time t in hours Your equation should contain no undetermined constants D What s the population of the bacteria culture at the end of 10 hours After 100 hours

1 Consider the curve y f x 2 1 A Find the exact area of the region in the first quadrant bounded by the curves y f x 2x and y x Exact area means no calculator numbers

3 A culture of bacteria is growing at a rate of 4e0 5 per hour within hours and 0 1 20 A How many new bacteria will be in the culture after the first six hours

Find a value of c 1 so that the average value of f x the interval 1 c is 0 9 A c 1 OB c 4 OC C 4 OD c 6 O E C 6 Cos 18

hind the Invense function show Laplace Thansform as of Variable to fully simplity as all your work F s S 5 d 3 3 2s 2 16

he formal integral Calculus definition Thans form to find the of the and Use of the Laplace Laplace Show all woonk FH eat Transform 1 following funct all propen nota

Consider the cost function C x 5x 14x 16 thousand dollars a What is the marginal cost at production level x 4 b Use the marginal cost at x 4 to estimate the cost of producing 4 25 units c Let R x x 51x 51 denote the revenue in thousands of dollars generated from the production of x units What is the break even point Recall that the break even point is when the revenue is equal to the cost d Compute and compare the marginal revenue and marginal cost at the break even point Should the company increase production beyond the break even point a The marginal cost at production level x 4 is

3 Let F x 3r cos 1 2 dt A Using the First Fundamental Theorem of Calculus find F x Yes this is as easy as it seems B Starting with F x f 31 cos 1 2 dr use the substitution u 1 to rewrite the definite integral You should get a new equivalent expression for F x which consists of this new definite integral

Write the slope intercept equation for the line having the given information 1 3 and 2 3 on the line What is the slope intercept equation of the line Use integers or fractions for any numbers in the equation

Find the slope of the graph of y f x at the designated point f x 2x 6x 2 3 2 The slope of the graph of y f x at 3 2 is

1 Practice finding the derivatives of functions that are defined by definite integrals 53 A Let g u 3 2 x dx Then d du B Find 3x 1 dt 20 cos 3xx

Find the area between the curve y interval 4 x 4 You get O A 184 OB 16 O c 212 OD 24 40 OE 3 2 and y x 3 on the

The region R is the region in the first quadrant bounded by the curves y x 4x 4 2 0 and x 2 as pictured below Find a value h such that the vertical line x h divides the region Rinto two regions of equal area You get h OA 0 44123 OB 0 42739 C 0 39987 OD 0 43120 OE 0 41260

The area of the region bounded by the curves y f x 2x 6x 2x 6 and y g x x 3x x 3 is given by A OB 3a 3x x 3 da 3x 3x x 3 da 3a 9x 3x 9 da 3x 92 3x 9 da L 3x 9x 3a 9 da 32 9x 3x 9 da 32 9x x 3 da 3 3a 9a x 3 da E 3x 9x 3x 9 dx

The definite integral of f x from a to b is defined to be b A So f x dx limn n Wh xk f a cb B f f x dx limn k 1 wk A k f cb O c f f x dx F a F b D f f x dx limn h 1 Ax wk k 1 a b O E f f x dx F b F a

The velocity of a particle moving along a line is t 2t sin t meters per second Then the net change in position during the time interval 0 t 4 is No calculators OA 20 376 meters OB cos 16 meters O C 16 45 meters OD 21 4 meters O E 7 cos 16 meters

Suppose that F 12 What is the value of F given that k 9 0 x 10 Q 9 x 104 Q2 9 x 104 and r 7 x 10

a Find the Laplace transform F s L f t of the function f t sin wt defined on the interval t 0 F s L sin wt Hint Use a double angle trigonometric identity b For what values of s does the Laplace transform exist help inequalities help formulas

I dx 3x tan x 4 dx O A 3x 2 sec x C B 3x4 tan x 4 OB O c 6x2 tan x 4 D 6x 3 sec x C O E 3x2 tan x C

5 Suppose a certain object moves in a straight line with the following velocity where is in meters per second and is in seconds v 2 1 3 sin t Without using your calculator but instead using properties of definite integrals and facts you know about area determine the net change in distance of the object from time 1 0 to time 6 seconds and find the object s average velocity on this interval

1 If r t is the rate of growth of a certain tree in feet per year where is in years what does the 40 expression r di represent

2 A population of rabbits starts off at time 0 with 225 rabbits and increases at a rate of r 1 rabbits per month where is in months 36 A What does the expression 3rd represent B What does the expression 225 r t di represent Suppose you re told that 24 425 How many rabbits are there in the population

4 Below is a graph of the function f x 5 4 3 2 1 It 4 5 6 7 y f x For what value of 0 is the average value of f x on the interval 5 c equal to 0 Exp

3 The marginal cost for producing x units of a certain product is given by the formula MC 0 001x20 5x 66 5 A What is the increase in the cost of the production if the production level is raised from 200 to 400 units You may use your calculator here Hint Remember that marginal cost is the rate of change of the cost of production

Use your calculator to estimate sin 5 a 2241 dx you get OA 0 991 OB 0 253 OC 0 495 OD 1 236 OE 0 784

47 Use your calculator to estimate f sin x 2 dx OA 3 628 OB 40 078 OC 4 734 OD 7 875 OE 4 320

Use your calculator to estimate to three decimal places 3 2 cos x dx OA 0 282 B 6 919 C 3 736 D 3 459 E 0 564

O A 1 000 OB 0 693 OC 2 000 OD 0 632 OE 2 914

Suppose that f x is continuous and that 5 x dx 3 x dx 2 and 3 x dx 1 Then 3 3 3 f x dx OA 5 OB 6 OC 3 OD 6 OE 3

The graph of y x 4 5x5 10x4 5x 10 is concave upward for all values of x such that A 0 x 4 B x 4 or x 1 C 4 x 1 D x 0 or x 4 E all real x

4 Consider g x x x 2 if x 2n 5 A Find the values of a and b such that g x is a differentiable function 12 points to B Write the equation of the tangent line to g x at x 2 9 points

Use the notion of signed area to evaluate the definite integral 2 x dx PROTIGHE 6 You get fo 2 x dx OA 8 OB 4 O C 12 OD 6 OE 4

Consider the function f x The graph is below f Use the notion of signed area to evaluate the following definite integral f x dx OA 4 TT OB 2 2T OC 4 4T OD 2 4TT OE 4 2TT 4 4 if 2 x 2 x 2 if 2 x 4