Definite Integrals Questions and Answers

You may use the following Constant Functions kdx kx C Exponential Functions e dx e C Trig Functions cos x dx sin x C csc x dx cot x C Inverse Trig Functions 21 6 21 1 Antiderivative Facts f x dx f x 4 f x dx S H S sin x dx cos x C sec x tan x dx sec x C arctan x C S de x x 1 Powers of x xdx dx arcsec x C P l p 1 21 You know that f x g x 5 and that You also know that f 6 3 and f 21 5 The region is bounded by the curves y f x y g x x 6 and x 21 The area of R is f t dt 28 1 C p 1 x dx In x C sec x dx tan x C csc x cot x dx csc x C dx arcsin x C

If f x is continuous and f f x dx 53 2 what is b xf x dx 0 Round your answer to two decimal places Answer

Let u 4 5 and v 2 3 Which of the following is equal to u v Select one a 7 b 7 O c 7 Od 23 23 e

Explain why no maxima or minima exist Minimize f x y 24x 6y subject to 6x 2y 80 Choose the best explanation below for why no maxima or minima exis A The value of is 0 B Fx 0 and Fy 0 have no simultaneous solution OC Fx does not exist D The constraint 6x 2y 80 is never satisfied

The graph of g is shown Let a 6 a Estimate b Estimate 18 1090 g x dx with six subintervals using right endpoints 18 18 c Estimate 18 a 18 100 18 0 a 9 g x dx with six subintervals using left endpoints g x dx with six subintervals using midpoints

Find fxx x y fxy X Y fyx x y and fyy x y for the function f f x y 9xe7xy f X V

Given that 1 5 21x x 4 dx 35 5 56 what is 21u u 4 du

a Estimate the area under the graph of f x 3 2x2 from x 1 to x 2 using three rectangles and right endpoints R3 Then improve your estimate by using six rectangles R6

ne speed of a runner increased steadily during the first three seconds of a race Her speed at half second intervals is given in the table Find wwer and upper estimates for the distance that she traveled during these three seconds ft smaller value ft larger value 0 0 5 1 0 1 5 2 0 2 5 3 0 v ft s 0 6 7 9 2 14 1 17 5 19 8 20 2

a Estimate the area under the graph of f x 4 x from x 0 to x 4 using four approximating rectangles and right endpoints Round your answers to four decimal places R

Find 9 So To f 0 C Enter f x dx if f x 5 for x 5 for x 5

Use the Midpoint Rule with the given value of n to approximate the integral Round the answer to four decimal places sin x sin x dx n 5 Ms

Write as a single integral in the form f x dx 3 2 f x dx f x dx b F x f x dx 1 r f x dx

9 So f x dx 33 and 9 I 2 g x dx 18 find 2f x 3g x dx

The graph of f is shown Evaluate each integral by interpreting it in terms of areas a b y 4 20 0 1 20 F f x dx 1 50 f x dx 1919 70 f x dx 90 20 y f x 40 60 80 X

A table of values of an increasing function f is shown Use the table to find lower and upper estimates for lower estimate upper estimate X f x 10 14 18 22 26 30 15 8 2 2 5 7 30 1 0 F X

Graph y 3 5 4 3 2 1 Clear All Draw 5 4 3 2 1 1 2 3 4 5 1 2 3 4 5

Convert to the indicated metric u Convert 50 in2 to square centime 0in cm

For what values of p is the integral convergent O a This integral is divergent for all values of p O b All values of p less than or equal to 1 O c All values of p greater than or equal to 1 O d All values of p greater than 1 O e All values of p less than 1 1 00 XP da

3a If f x is continuous and f f x dx 485 1 what is Round your answer to two decimal places Answer f 3x da

Use a change of variables or the table to evaluate the following definite integra KIN KIN 2 S 2 cotx csc x dx 4 Click to view the table of general integration formulas 2 S 2 cotx csc x dx Type an exact answer

Use a substitution of the form u ax b to evaluate the indefinite integral below 4x 3 dx 4x 3 dx

Find the net area and the area of the region bounded by y T between x question OA and x Graph the function and find the region indicated in this 2 The net area is Simplify your answer The area of the bounded region is Simplify your answer Graph the function y 11cosx and find the region bounded by the function and T the x axis between x and x Choose the correct graph with shaded region 2 indicated below Ay 2 3 0 OB 27130 O C AV 23 0 O D 2340

Simplify the following expression 3 d dp dx S 2 7 P tu dp C

Use the Fundamental Theorem of Calculus to evaluate the following definite integr 4 511 dt 14 0 5 dt

Suppose f x dx 6 f x dx 8 and g x dx 4 Evaluate the following integrals 2 7 2 7 g x dx 7 Simplify your answer 2 4g x dx 2 Simplify your answer g x f x dx 7 7 f g x 2 Simplify your answer 17g x f x dx 2 Simplify your answer 2

2 LO 3 Given the table of values below use a Right Riemann sum with n 4 to estimate the area bound by the curve on the interval 3 11 Show your work by writing out the expression you are calculating X f x 3 2 4 3 5 5 5 6 2 7 1 8 0 9 3 10 1 11 4

3 2 pts Properties of definite integrals Suppose that we know the values of the following integrals 3 f x dx 2 f x dx 3 Find the value of f x g x dx g x dx 1 g x dx 3

4 2 pts Definite integrals Compute the following definite integral 1 31 dx

Determine which figure shows supplementary angle a d A 2 b e 1 2

a To find the values of that correspond to the pole you need only look at the graph of the curve and determine the values by visual inspection true or false b This problem is in the category of hard case or case 2 of finding the area bounded by one curve where the region involves the pole Find the area of the region 2 inside the smaller loop of 2sin 0 the curve r 1 5n This Q 11mg

f x xsinx 2 x 6 n 5 Round your answer to two decimal places

late fe sin y dx e cos y dy where C is any curve connecting 0 0 and 1 5

se the method of cylindrical shells to find the volume generated by rotating the region bounded by the graphs of the given equations about the y axi y 5x y 30x 10x 5T 2 X

ard my cours ne Assignments Assignment 2 PC MATET

In the following exercises use a change of variables to evaluate the definite integral 1 x 1 2 da 292 So 0 293 294 2 X 1 x 2 dx t dt 5 t

Evaluate the definite integral 3 x 3 In x O 9 863 O 1 140 O7 731 dx

atch the variable to its corresponding value 2 10 1 2 4 x 2 y 8 Z z w 3

VIDEO 7 nments Assignment 2 If f x is continuo

Let the region R be the area enclosed by the function f x x 1 and g x 2x 1 Find the volume of the solid generated when the region Ris revolved about the line y 4 You may use a calculator and round to the nearest thousandth 15 1 0 5 2 8 10 05 1 15 2

Evaluate the given definite integral using the fundamental theorem of calculus 1 4 el x 2 1 6 dx e6 e4 e4 e6 Oel 6 el 4 1 4

question 1 Find the indefinite integral sin x S FR as possible dx 3 cos x

A child pulls a wagon with a force of 40 pounds The handle of the wagon makes an angle of 30 with the ground Express the force vector F in terms of i and j

Finding I in spherical coordinates Find the moment of inertia about the z axis of a solid of constant density 8 1 that is bound ed above by the sphere p 2 and below by the cone TT 3 spherical coordinates

signments Assignment 4 of A probability density function for the ran a constant What value must k be if X tak if 0 x 1 r 1 r 1 r

Find the exact length of the polar curve oses 102 el r 2 sin 8 0 es

Evaluate the following integral in cylindrical 21 5 5 SS Saz dz r dr d0

Calculate the exact value of cos 5 using the half angle formula Rationalize the denominator if necessary Hint sin 5 2 cos 5 Enter a or sign

Use a double integral to find the volume of the indicated solid Z 6 Z 6 2y y

1 point If f is continuous and S f x dx 7 evaluate f 5x d