Definite Integrals Questions and Answers

Sketch the region enclosed by the given curves y cos x y sin 2x 0 x 1 2 y 2 7 0 16 Find the area of the region Need Help Read It KIM 2 2 00 2 6 3 O y 1 7 6 IN 00 2

Sketch the region enclosed by the given curves y x 2 x y 2 y 2 1 1 6 2 1 Find the area of the region 3 y 2 00 2 1 2 002 1 2

Assume x and y are functions of t Evaluate dy dt dx x y 9 3 x 1 y 2 dt OA OB OC 3 C 3 3

ASK YOU Set up but do not evaluate an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line y 8x x y x about x 10 dx

The following integral represents the volume of a solid of revolution Describe the solid 5 y dy The integral 2 Y Y x x y dy xy plane about the y axis dy describes the volume of the solid obtained by rotating the region x y Osy 4 R 0 x 16

Evaluate 3 6x O 163 5 O 160 O 146 O 142 5 21x 5 dx

The graph shown is f x x 3 What is the integral function g t that defines the area under this line between x 3 and x 5 54321 1 2 3 4 5 2 3 4 Og t t 3 dt Og t 3t dt 9 g t 3 t 3 dt 5 g t 2 3 1 3 dt

Consider the function f x x x 1 Select all that apply Select 4 correct answer s Of has a horizontal asymptote at y 1 f has a horizontal asymptote at y 1 f has a vertical asymptote at x 1 f has a vertical asymptote at x 1 The limit of f x as x approaches infinity is 1 The limit of f x as x approaches infinity is 1 The limit of f x as x approaches negative infinity is 1 The limit of f x as x approaches negative infinity is 1

Three regions are defined in the figure C 0 5 y 0 R y 5 x R3 y 5x R B 1 5 A 1 0 Find the volume generated by rotating the given region about the specified line about OA

Sketch the region enclosed by the given curves y 12x y y 4 6 y 3 X 0 6 8 2 4 6 DO MY NOTES ASK YOUR TEACHER 3 4 X PR

The graphs of two functions are shown with the areas of the regions between the curves indicated YA 0 12 2 b What is the value of 3 25 4 9 F x g x dx 5 a What is the total area between the curves for 0 x 5 X

Solve the following equation for 0 S X sin x cos 2x Evaluate the integral and interpret it as the area of a region 2 1 2 Sketch the region y 22 12 sin x 2 cos 2x dx 4 DO DO

Sketch the region enclosed by the given curves y 14 x y x 4 6 6 15 10 y 5 y 5 5 10 15 Find the area of the region 6 DO 6 4 2 S 10 y 15 2 4 DO 6 2 MY NOTES 15 10 2 ASK YOUR TEACHER

Find the area of the shaded region A y x 2 4 2 8 6 2 1 6 3y 2x 16 2 2 4 y 8 2x 4

Evaluate the definite integral n 4 4 x x8 tan x dx

Evaluate the definite integral 36 dx xv In x

Evaluate the definite integral ex bora 0 1 e x dx

Evaluate the definite integral el x x dx

Find the area of the region bounded above by y x 8 and below by y x O O O 32 58 3 64 3

Decide whether the integral is improper 8 10 St sin x dx 57 x O proper improper Explain your reasoning Select all that apply The limits of integration are both finite The integrand is continuous on The integrand is not continuous on 0 At least one of the limits of integration is not finite

Question Determine the end behavior of the polynomial functions below 1 f x 3x7 4x6 5 Select 2 g x 3x7 Select 3 h x 4 Select Soloct 1 4x 3 X6 5 4 j x 3 8x 2x 15x x

4 Utre State the domain and range for the given graph of a parabola 8 7 6 5 4 3 2 1 DD The domain is 654921 The range is 2 3 1 0 0 2 0 Empty Set 3 4 00 4 4 00 5 7 1 6 0 4 7 2 41

What formula would be used to calculate the right rectangle approximation of 2 sin x cos x dx using four subintervals O R4 sin 0 cos 0 sin cos sin r cos r sin 3 cos 3 O R4 sin cos sin z cos z sin 3 cos 37 sin 27 cos 2 O R4 sin 4 cos 4 sin 3 cos 3 sin 5 cos 5 sin cos O R4 sin 0 cos 0 sin cos sin z cos z sin 3 cos 3 21 22 23 24 25 26 27 28 29 30 Next

Find the area under the graph of y x 4x on the interval 3 1 5 4 3 2 1 1 2 3 4 5 2 3 OO O 16 3 22 3 9

The base of a three dimensional figure is bound by the line y x on the interval 0 2 Vertical cross sections that are perpendicular to the x axis are rectangles with height equal to 4 Find the volume of the figure H 3 2 11234567 O O O 5432 55 23 23 44 3 O 48 32

Question 5 4 3 2 1 9 4 5 The x intercept with a positive x value Use the graph to identify the points for the described intercepts below The x intercept with a negative x value The y intercept with a negative y value 1 The y intercept with a positive y value 2 3 4 5 4 1 0 4 2 4 0 3 0 3 4 2 0 5 3 0 6 3 2 7 3 0 8 0 2

If the tangent line to y f x at 6 5 passes through the point 0 4 find f 6 and f 6 f 6 f 6

Given that f x dx 53 13 2 answer 2 5 what is f s ds I Question Help Video Message instructor

a 12 s ESPRES ELIF BA f t If f t

1 C 4 3 2 1 a To find the blue shaded area above we would calculate f x dx area Where 1 2 3 4 a f x area b Question Help DVideo Message instructor

Given cos 0 find sin 0 8 17 sin 0 07 Simplify your answer if possible P

8 Find in degrees 17 0 degrees Round to the nearest hundredth Ente

3 Determine whether the following improper integrals converge or diverge If it converges specify the value it converges to a T b d T Hint Use integration by parts and take for granted the fact that lime te t 0 We will learn how to evaluate a limit like this next week or so e f dr redr I 2 1 dr dr 1 x Hint Note here that the integrand has a discontinuity at z 1 Use trigonometric substi tution to find the antiderivative it is case 1 from the lecture 4 notes Finally recall that arcsin 1 and arcsin 0 0

Connecting Average Value and Net Signed Area 20 15 10 Average value 6 Not enough information O 52T 211 102 367 3m 4 1 411 The figure above shows a function f x on the domain 0 x 6 On this interval the function has average value 6 What is the value of f f x dx the net signed area trapped between the function s graph and the x axis on 0 x 6 x 511 611

9 0 89 Points Evaluate the piecewise defined function at the indicated values x f x x 3 f 2 f 1 f 0 f 1 f 2 DETAILS if x 0 if x 0

A region bounded by f y y y 1 y 4 and x 0 is shown below Find the volume of the solid formed by revolving the region about the y axis 5 2 3 221 O 2 3 1234 567 27 O 13 T o 21n

For y f x 7x find Ax Ay and Ax Ay Ay Ax Ay Ax given x 1 and X 4

Evaluate O O 0 3 cos 2x dx 2 3 12 3 O 4 28 29

1 25 points Evaluate 1 1 tan 2t dt lim 1 0 I Jo 2 25 points Determine the area of the region bounded by x y y 6 and x 2y 4 3 25 points Evaluate the following integral In 2 1 de 1 4 25 points Find the volume of the solid obtained if the plane region R which is bounded by y 1 sin c and y 1 from z 0 to x is rotated about the z axis

Find the volume of the solid bounded by the sphere x y x 2x x 1 and th cone x y z Evaluate by using cylindrical coordinates

Find f g f g fg and fig g x domain x 2 2 9 x x27 x f g x domain fg x domain 12x 35 x x 7 0 7 U 7 0 U 0 00 35 2x x x 7 0 7 U 7 0 U 0 0 35 x 7 0 7 U 7 0 U 0 00 5 5 4x 7 Your aRCHHOE can More Informati

12 50 Puanlar 1 The line y x and the parabola curve y x divide the xy plane in five areas of which one is bounded Denote this bounded area with R and calculate the double integral SSR x y dA a b c d e 4 5 1 15 9 2 7 12 3 20 Bos b rak

4 a Find the volume beneath the surface f x y 3x 2y over the region R given in the Figure b Let 1 2 Lore Love J xy dy dx 1 1 Apre 15p Sketch the region of integration and write an equivalent integral with the order of integration reversed X

Let f be a twice differentiable function on lim f x 2 8118 lim f x 8118 lim x f x 5 8118 then evaluate x x da f dx O A 8 OB 4 C 2 OD 2 E 6 0 with f 0 1 If the following limit results are provided

For a 0 let f x be an integrable function given with f 3 0 f 2 a and f 6 4 Find the value of a if we k that 2x f 2x f 2x dx 3 O A 1 IN OB 3 312 O 03 OD 2 512 O E5

we the initial boundary value problem for the given heat equation u 3u 0 x 2n t 0 u 0 1 u 2 t 0 t 0 u x 0 8sin 3x 5 sin 4x 2sin 7x 0 x 2

Find the area 22 ft 138 Br 38 09 ft C 109 A D 37 ft square feet Round to the nearest hundred

Find the volume of the region D bounded above by the spher x y z 360 and below by the paraboloid 2z x y

1 a Use the formula to find the length of the indicated portion of the curve r t t sint cost i t cost sint j 4k 0 t n L B dt 2 dz dt 2 dt 15p

4 For the following problem a Sketch the region bounded by y x y x b Find the area of the bounded region