Definite Integrals Questions and Answers

If you know that f x f x f x and that f 9 9 while f 7 9e then 7 2 f x f x dx

The exact average value of f x x In x on the interval 15 e e is

If 4 4 4 14 3 10 and f 3 5 the 3 1 x f x dx

y 1 x y 0 x 1 FEIN QUESTION 5 Find the volume generated by revolving the region bounded by the given curves about the y axis Use the method of disks y 4 x2 y 0 x 0 x 2 6m 8n 50 n

Tanisha Vizquez owns a wallpaper hanging busine the number of square feet of wallpaper she hangs

Solve using the zero factor proper 11c 5 6c 18 0

53 59 The region bounded by the given curves is rotated about the specified axis Find the volume of the resulting solid by any method 53 y x 6x 8 y 0 54 y x 6x 8 y 0 55 y x 1 y 2 56 y2x 1 y 2 57 x y 1 1 about the y axis about the x axis about the x axis about the y axis about the y axis

39 42 Each integral represents the volume of a solid Describe the solid 3 39 27 2 x dx

10 points Examine the existence of the following improper integrals a 5 points b 5 points 63 tan x dx p x 1 x X dx

9 14 Use the method of cylindrical shells to find the volum generated by rotating the region bounded by the given curve about the y axis 9 y x y 0 x 4

4 Knowing that 1 1 x x x 1 20 approximate with a tolerance level of 0 0001 the following definite integra 0 2 X 1 z dx

1 Solve the a u du x at au boundary value problem x 0 1 0 100 lim u x t 0 t 0 1148 axlx 0 u x 0 0 x 0 using the Laplace transform Give two different forms of t solution u x 1

24 Given tane that satisfies the given conditions What is the correct graph A B C D 12 5 8 12 and is in Quadrant IV Sketch an angle in standard position N 12 12 5 5 I

14 Without using your calculator evaluate the expression sec 30 cot 60 A 1513 B 11 C 3 D 1

Given the following acceleration function of an object moving along a line find the position function with the given initial velocity and position a t 0 4t v 0 0 s 0 8 s t Type an expression using t as the variable

K Graph the equation using the s 5x 1x 16

a c 1 TIMER 3 TH Evaluate sin x cos xdx FASTE What 761 1 sin x sin x C 5 3 5 1 3 1 sin x sin x C 5 8 b 1 3 1 cos x cos x C 5 d cos 1 3 1 cos x cos x C

5 sin x 1 2cosx dx

Evaluate the integral 00 2 1 S 3 x 1 dx

00 1 x lnx 3 de

Factor the trinomial 2x 2 9x 9

f x x 2 10 What is the average rate of change of f over the in

3 10 Finding an Angle or Side Use the Law of Cosines to com determine the indicated side x or angle 0 C 4 C 3 A 21 39 42 X B 15 A 108 18 B

51 Height of a Tower A water tower is located 325 ft from a building see the figure From a window in the building an observer notes that the angle of elevation to the top of the tower is 39 and that the angle of depression to the bottom of the tower is 25 How tall is the tower How high is the window 39 25 325 ft WATER VAVAWA

3 Use the Rational Zero Theorem to list all the possible rational zeroes of f x 3x 2x 16x 24

2 Is 4 a lower bound for the function below How can you tell x 5x 10x 12x 20

For the following exercises perform the indicated operation m 4m 21 5m 10m m 9m 20 2m 10 55 10m 80m 3m 9

Let ABC be a triangle Show that E LA B Zc Sin a cos a

You know that 11 1 th 96 5 g t dt 27 4 3 g 11 4 3 g 5 Evaluate 5 11 g x 16 a r dx

In Problems 51 56 use modal decomposition to solve 23 8 8 0 51 A X 52 A 1 1 2 2 orn 0 Ax f x 0 xo 0 xo 1 X x1 2 Xo

y dx Here y f x and with values given in the table e Simpson s Rule and all the data in the following table to estimate the value of the integral 2345678 JT 9 4 123710

ref L x be the reflection about the line in L R shown in the accompanying figure a Draw sketches to illustrate that T is linear b Find the matrix of T in terms of

6 5 x a Find the slope of the line tangent to its inverse function at the point P 1 1 For the function y f x m b Find the equation of the line tangent to the inverse function at the point P 1 1 y

The following is the graph of a function f x 5 4 3 2 1 Clear All Draw 4 3 2 1 1 2 3 4 5 7 21 3 4 a On the same axes sketch a graph of the inverse function f x b Use the graph of the inverse to evaluate f 2

Marcello s Pizza charg s a base price of 11 for a large pizza plus 2 50 for each additional topping a Find a function that models the price of a pizza with n toppings b Find the inverse of the function f f p What does f1 represent Of 1 represents the number of pizzas with toppings that cost p dollars Of 1 represents the cost of a pizza with n toppings O f1 represents the number of toppings on a pizza that costs 2 5 p dollars O f 1 represents the number of toppings on a pizza that costs p dollars Of 1 represents the cost of n toppings on a pizza c If a pizza costs 16 how many toppings does it have toppings

Use the Table of Integrals in the back cover of Stewart s text to evaluate the integral 6 si Note Use an upper case C for the constant of integration 6 sin 2x dx

Evaluate the integral by changing the order of integration in an appropriate way 00 35 7 SS Se x 2 dx dy dz 0 0 y 5 O A 1 49 5 O B 1 e 49 5 OC 1 e 56 5 56 C

Evaluate the definite integral 5 0 10 dx

and 0 0 0 1 2 0 bounds a region R in the plane Calculate the volume bound between the graph of the surface 2 x y and the region R in the xy plane

Find the value of 3 2 3 2 e coda sin q dq 0

R a b show that f f x dx lime a f f x dx

The graph of f is shown below Evaluate each integral by interpreting it in terms of areas 1 f f z da 1 dx Next Problem 2 f x dx 2 2 3 1 2 dx Note You can click on the graph to enlarge the image

1 point 13 You are given the four points in the plane A 3 4 B 1 1 C 4 3 and D 8 3 The graph of the function f z consists of the three line segments AB BC and CD Find the integral f a da by interpreting the integral in terms of sums and or differences of areas of 3 elementary figures 1 1 2 dz 0 f x

5 and 2 80 Find a formula for the exponential function passing through the points 2 16

Use the Table of Integrals in the back of your textbook to evaluate Su Perform the substitution U 27 6y y dy Use formula number int y sqrt 27 6 y y 2 dy

2 4 T Find the length of the curve y In cos x x 0

Urban Community College is planning to offer courses in Finitel Math Applied Calculus and Computer Methods Each section of Finite Math has 40 students and earns the college 40 000 in revenue Each section of Applied Calculus has 40 students and earns the college 60 000 while each section of Computer Methods has 10 students and earns the college 26 000 Assuming the college wishes to offer a total of seven sections accommodate 220 students and bring in 292 000 in revenues how many sections of each course should it offer Finite Math section s Applied Calculus section s Computer Methods

4x 2y 5pz 6 4x 6y 5p 3 z 6 q 2xy 2pz 2 1 A system of equations is given a Use Gaussian Elimination to rewrite the system in row echelon form b Hence state the conditions for p and q where there would be infinitely many solutions c Hence state the conditions for p and q where there would be no solutions

Aaron invested 740 in an account paying an interest rate of 5 1 2 compounded continuously Justin invested 740 in an account paying an interest rate of 5 7 8 compounded daily After 18 years how much more money would Justin have in his account than Aaron to the nearest dollar

Use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x axis Round your answer to four decimal places y e y 0 X 0 X 3