Definite Integrals Questions and Answers

oint Suppose luate the definite integral by interpreting it in terms of signed area 0 f x dx f x 3x 15 15 3c if 0 5 if 5 10 gestion Draw a picture of the region whose signed area is represented by the integral Then find the signed area using formulas from high school geometr

If q s q s L VA 8 cos v v dv then

Let A x 3 2 1 1 2 3 4 T A 2 A 2 I f t dt where f is the function given by the graph below 3 4 6 7 8 q f 4

How large should be to guarantee that the Trapezoidal Rule approximation to 1x 1 dr is 2 22 122 accurate to within 0 0001 n How large should n be to guarantee that the Simpsons Rule approximation to 2 27 127 17 1 dr is accurate to within 0 0001 n Hint Remember your answers should be whole numbers and Simpson s Rule requires even values for n Question Help Video 1 D Video 2 Message instructor

Use Simpson s Rule and all the data in the following table to estimate the value of the integral 1 10 F yda on 10 6 1 N N 4 Pa 0

Use Simpson s rule with n 6 to approximate cos z dz Keen at least 2 decimal places accuracy in your final answer

The following table shows the rate of water flow in L min through a dam t min 1 6 V t L min 6 1 5 1 16 21 26 31 15 liters 69 5 Approximate the total volume of water that passed through the dam from t 1 tot 31 using Simpson s rule with n2 6

Use the Midpoint Rule with n 3 to approximate the integral 2x 102 da

A region bounded by f y y 4 y 1 and x 1 is shown below Find the volume of the solid formed by revolving the region about the y axis 3 2 1 5432 234 2 3 88 T 3 20n O 18n O 20 3 ob wit Nw T

Which integral would represent the area shown The function is f x x 5 3 2 0 0 5 1 1 5 2 2 5 Created with a trial version of Advan O 2 24 x 5 dx 5 x dx 5 x 5 dx o f 5 x 5 dx O

What is the Riemann sum that represents the illustration shown The function is f x x 5 o xi i 1 O O O xi Axi i 1 3 xi i 1 xi Axi i 1

Evaluate O 24 O 30 O 36 O 42 2x 7 dx 25 26 27 28 29

What formula would be used to calculate the left rectangle approximation of 2x 2 dx using five subintervals O L5 0 4 2 0 2 2 2 0 6 2 2 1 0 2 2 1 4 2 2 1 8 2 O L5 0 4 2 0 0 2 2 0 4 2 2 0 8 2 2 1 2 2 2 1 6 2 O L5 0 4 2 0 4 2 2 0 8 2 2 1 2 2 2 1 6 2 2 2 0 2 O L5 0 2 2 0 2 2 2 0 6 2 2 1 0 2 2 1 4 2 2 1 8 2 A 20 21 22 23 24 25 26 27 28 29

Find the area of the region bounded to the right by X y 4 and to the left by x y 2 R2 0 O 4 5 04 2 4 6 8 O 3 75 O 8 3 61 62 63 GA GE 66 67

A region bounded by f y 2y 2 y 0 y 3 and x 1 is shown below Find the volume of the solid formed by revolving the region about the y axis 10 4234 2 3 4 5 6 7 8 9 O 36 O 63 O 242 T O 817

Rewrite the integral using substitution x cos x dx 25 5 cosu du O O O 25 5 cosu du 125 cosu du 125 cos u du 30 31 32 33 34 35

An object moves along a path with a velocity v t 1 t and the initial position of the object is x 0 2 What is the equation for the position of the object when the acceleration is zero 0 2 u u du 0 2 u du 0 2 u u du lo u u du 48 49 50 51 52 53 54 55 56

Rewrite the integral using substitution O O O u du 68 10 1308 u du 68 u 0 du 64 u du 3 10 x x 4 0 dx 29 30 31 32 33 34

Find the indefinite integral Use C for the constant of integration Jar arcsec 3x dx x x arcsec 3x 1 w r In 9x 1 3x 2 3 C

Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y axis y x 2 I 2

Calculator Permitted If the midpoints of 4 equal width rectangles is used to approximate the are nclosed between the x axis and the graph of y 4x x the approximation is A 10 B 10 5 C 10 666 D 10 75 E 11

The equation r t 3t 9 i 5t j t2 k is the position of a particle in space at time t Find the angle between the velocity and acceleration vectors at time t 0 The angle is radians Type an exact answer using as needed

Evaluate f c at the point f x e x 3 c 1 3 O A f 1 36 OB f 1 6 OC f 1 42 OD f 1 1

Evaluate the difference quotient for the given function Rationalize the numerator and simplify your answer f x f 1 x 1 f x x 4

Jo 1 y dz dy dx over the region D is given D is a bounded region in space given in the Figure below Write an equivalent integral in order dydzdx and find the value of the integral Side y r Top y z 1 1 1 1 0

4 23 Find the value of ze at the point 1 1 0 where yz x y e

Sketch the region of integration for the following integral and evaluate the integral 2 2 x xe y 12 2 Y 0 0 dy dx

What is the mass of the space object bounded by the cylinder x y 2x and the cone z x y2 if the density function is p x y z 4 x y O A 3 8 OB 972 8 O C 1572 8 O D 37 2 OE 372 4

2 Find the area of the region bounded by the graph of f and the x axis on the given interval f x 9 x on 2 3

2 1 1 2 0 2 f 3 4 decreases as x increases range of 5 5 Match each transformation of function f with a feature of the transformed function X domain of 5 j x f x 5 h x f x g x f x 5 x intercept of 1 0

5T Use a half angle formula to find the exact value of cos 8 cos 5n 8 0 8 10

a Find an approximation to the integral using a Riemann sum with right endpoints and n 8 Rg 1 5 b Draw a diagram to illustrate the approximation in part a y y 5r 3 2 1 2 0 31 4 3 2 1 3x dx 2 O 34 1 2 1 3 2 2 5 2 3 c Consider the following theorem 2 7 2 Use this to evaluate the integral 4 4 If fis integrable on a b then 4 2 9 5 5 5 4 3 2 4 3 2 2 0 3 f x dx f x dx lim x Ax where Ax n l 3 2 d Interpret the integral in part c as a difference of areas The integral represents the area below the x axis minus the area above the x axis O The Integral represents the area above the x axis minus the area below the x axis 4 9 9 5 x b a and x a AX n

Let g x fr f t dt where f is the function whose graph is shown f VA 5 10 15 20 25 30 35 g 10 g 15 g 20 g 25 g 30 15 10 a Evaluate g x for x 0 5 10 15 20 25 and 30 g 0 g 5 S 5 t b Estimate g 35 Use the midpoint to get the most precise estimate g 35 X c Where does g have a maximum value Where does it have a minimum value minimum X maximum d Sketch a rough graph of

Evaluate the definite integral 11 13 0 3x 3x dx

Consider the following a Find an approximation to the integral using a Riemann sum with right endpoints and n 8 Rg b Draw a diagram to illustrate the approximation in part a y Sr 5 y 4 3 1 2 2 x 3x dx O 3 7 2 c Consider the following theorem 4 If f is integrable on a b then Use this to evaluate the integral 5 4 2 2 0 3L 2 2 n x f x dx lim f x Ax where Ax n 1 1 5 b a n 2 2 0 3 1 and x a Ax 2 5 3 7 NI 2 4 9 5 X 3 2 2 20 34 3

Which of the following is a Riemann sum corresponding to the integral 2 x4 dx X4 A r 2 B 2n 1 C 2 D 1 n E 7 1

Soru 10 Puan Which of the following is a Riemann sum corresponding to the integral 2 x4 dx A E 2 B 2 c i 1 Li 1 E 9 O A B D 1 i 1 n

Puan Let R be the region bounded by the curve y x the line y 1 and the y axis Finc the volume of the solid generated when R is revolved about the line y 1 7 17 P 3 17 E TT y 1 y Vr D TU E F

2 Find the general solution of the following system in parametric form by using Gaussian elimination method x 3y 5z 1 2x 4y 6z 2 3x 7y 11z 3 3 Find the dimension and a basis for the

1 e 4 14 In4 4 f x y dydx f x y dydx f x y dydx 3 1 x 1 e e 1 ex 4 ln y ex ex f x y dxdy B f x y dxdy C f x y dxdy D f x y dxdy E f x e 1 y 0e 3 1 y 3 e

15 lim 848 E n 1 3 n 1 1 2 3 n 2 1 1 3 3 n 1 1 n 3 n 2

In Problems 14 15 evaluate each limit by interpreting it as definite a integral 2 14 lim 32 n 34 n 36 n 3 1 00 N 32n n

2 Find the total area enclosed between the large loops and small loops of the curve r 1 2 sin 30

1 of the questions remains unanswered 3 points Write a triple integral including limits of integration that gives the volume of the cap of the solid sphere x y2 z 17 cut off by the plane use theta rho and phi for 8 p and as needed 1 and restricted to the first octant In your integral What coordinates are you using Enter cartesian cylindrical or spherical With a C e Ja Se S Volume correct b d and f d d

If f x 2 cos x 0 x evaluate the left Riemann sum with n 6 taking the sample points to be left endpoints Find the width of each subinterval 3n 4 X1 x1 x2 Find the left endpoints of the subintervals where x x X6 3x 24 x3 24 X5 33 53 53 53 x6 24 12m 15m x 3a units x x x x 3x 4 x Evaluate the Riemann sum Round your answer to six decimal places 1 785164 X What does the Riemann sum represent The Riemann sum represents the net area of the rectangles with respect to the x axis Illustrate with a diagram 20 21 OX Q

Find T N and x for the plane curve r t 6cost 6t sin t i 6sin t 6t cos t j t 0 T t i i Type exact answers using radicals as needed

0 1 Points DETAILS Use the Midpoint Rule with the given value of n to approximate the integral Round the answer to four decimal places 70Vx M6 71 9996 X PREVIOUS ANSWERS dx n 6

The graph of a function g is shown b left endpoints 14 c midpoints 0 Estimate g x dx with six subintervals using the following a right endpoints N

3r If f x 2 cos x 0 x evaluate the left Riemann sum with n 6 taking the sample points to be left endpoints Find the width of each subinterval units Find the left endpoints of the subintervals where x x X6 x1 x2 x3 X4 5 x6 4 Evaluate the Riemann sum Round your answer to six decimal places What does the Riemann sum represent The Riemann sum represents the net area of the rectangles with respect to the Select O Illustrate with a diagram NEW 4 37 0 2 3X 0 21 3a 8 20 2 5m 31

The graph of f is shown a b y d y f x 6 K 0 6 12 18 Evaluate each integral by interpreting it in terms of areas f x dx 15 f x dx 0 f x dx F x dx 24 18x dx X