Definite Integrals Questions and Answers

Find the arc length of the graph of y 1 x 1 y 5 a 8 5621 c 9 0734 b d 32 8287 10 5832

Let C be the curve 27x y 0 between y 0 and y 2 Find the area of the surface generated by revolving C about the y axis Please set up do not evaluate a c 54 2 9 2 0 2 x 1 23 0 15 dx 81 dy 9 b d 2 2 27 2 2 STO 1 0 dy 81 dy 81

Recall that the graph of a single equation in three variables is a plane in 3D space Select all possibilities for the set of all solutions to a system of two linear equations in three va 2 0211 222 2 b There could be exactly two solutions There could be no solution t There could be a unique solution uld be infinitely many solutions

Write the first expression in terms of the second if the terminal point determined by t is in the given quadrant tan t cos t Quadrant III tan t

Find the value of each of the six trigonometric functions if it is defined at the given real number t Use your answers to complete the table If an answer is undefined enter UNDEFINE 3 2 t t 3A sin t cos t tan t csc t sec t cot t

5 Consider the definite integral fx tan x dx a Explain why this integral cannot be evaluated exactly by using either u substitution or by integrating by parts b Using appropriate subintervals compute L4 R4 M4 T4 and Sg c Which of the approximations in b is an over estimate to the true value o So x tan x dx Which is an under estimate How do you know

Find the area under the graph of f x tan 3 2 theta over the interval form 0 pi 6 Please demostrste it with the triangle pethagores theorems showing the ankles too

ork 1 3 1 4The Derivative of 0 16 answered on 1 hat f is a function given as f x 2x 5 will compute and simplify the expression f 4

Find the rate of change of the volume of a cube with respect to the length of its side s when s 17 Give your answer as a whole or exact number rate of change cubic units per unit incre

5 Determine a formula for the approximation curve using of the area under the x x 4x 4 over the interval 1 4 LN Then find the exact area by finding lim LN N

What does the central limit theorem allow us to assume about a set of data O All the values are positive O It follows a normal distribution It does not follow a normal distribution

What does the standard deviation indicate about a set of data The difference between the highest and lowest values The average variation of the data from the mean O The mathematical average of the data O The middle number in the dataset

The range of which function is 2 00 O y 2X O y 2 5x O y 5x 2 0000 y 5x 2

10 Points SCALCET9 15 XP 6 009 Evaluate the triple integral iiint E 2x d V text where E left x y z mid 0 leq y leq 2 0 leq x leq sqrt 4 y 2 0 leq z leq 4 y right

Reasoning and Proof list 1 2 K Identify the hypothesis and conclusion If the measure of an angle is exactly 90 Identify the hypothesis Choose the co 00 Part 2 of 2 A It is acute B The measure of an angle is g It is a right angle

Find the length of the arc of a circle of radius 6 inches subtended by a central angle of radians 5T 3 inches help numbers

10 5 pts Use the Squeeze Theorem to show lim x 3 sin 0 Make sure your work has specific x 0 logical steps

The graph of one complete period of a cosine curve is given wo 12 3 HUS horizontal shift 12 02 2x a Find the amplitude period and horizontal shift Assume the absolute value of the horizontal shift is less the the period amplitude period X b Write an equation that represents the curve in the form y a cos k x b

Let f t 2t 14 and consider the two area functions A x f t dt and F x f t dt Complete parts a c 10 a Evaluate A 8 and A 9 Then use geometry to find an expression for A x for all x 27 The value of A 8 is 1 Simplify your answer The value of A 9 is 4 Simplify your answer Use geometry to find an expression for A x when x 27 x 14x 49 Type an expression using x as the variable b Evaluate F 11 and F 12 Then use geometry to find an expression for F x for all x 2 10 The value of F 11 is 7 Simplify your answer The value of F 12 is Simplify your answer

The graph of f is given in the figure to the right Let A x f t dt and evaluate A 2 A 6 A 10 and A 14

Consider the function f x x 9x a Graph f on the interval x 20 b For what value b 0 is ff x dx 0 c In general for the function f x x ax a 0 for what value of b 0 as a function of a is a Choose the correct graph below 2009 40 b b Type an integer or decimal B 200 40 is ff x dx 07 0 Q OC 200 40 Q OD 2007 40

Consider the function f t 3t 2 and real number a 2 ff t dt for f a Find and graph the area function A x f t dt b Find A x a The area function is A x A x area

The region R in the first quadrant is bounded by y 4 sin z y 4 z 2 and y 3x 1 and contains the point 2 4 The area of R is area The volume of the solid generated by rotating R about the x axis is volume The volume of the solid generated by rotating R about the y axis is volume The volume of the solid with R as its base and square cross sections perpendicular to the 2 axis and to the xy plane is volume

You want to use the tube method to find the volume of the solid obtained by rotating about the y axis t region in the zy plane bounded by the curves y 102 and y 0 A typical tube has radius circumference 2x and height f 1 Using the tube method the volume is given by the integral S where a Evaluating the integral you found above volume da

A bucket has a mass of 10 kg when filled with sand You need lift it to the roof of a building 10 meters tall using a rope that has a mass of 0 2 kg m You need 1 meter of rope to secure the rope to the bucket Once the bucket reaches the top of the building it has mass 8 kg not including the rope because sand leaked out of a hole in the bucket at a constant rate while you were steadily lifting it to the top of the building The work done lifting the bucket sand and rope to the top of the building is Recall that accoloration dug Joules 0 91

A conical trough has a radius of 4 feet and a height of 9 feet It is filled with a liquid that weighs 22 pounds per cubic foot and the depth of that liquid is 7 feet Find the amount of work in foot pounds required to empty the trough by pumping the water over the top

Use the tube method also known as the method cylindrical shells to find the volume of the solid obtained by rotating the region bounded by y x y 0 and x 4 about the y axis volume

Using the given graph of the function f find the follo a The numbers if any at which f has a local maximum What are these local maximum valu b The numbers if any at which f has a local minimum What are these local minimum value

Suppose that f and g are continuous and different ALL of the following conditions are also true f x dx 3 g x For parts a e find the numerical value indica answer the question indicate Insufficient informa steps a h 5 3 f x dx 4 f 0 g x dx 0 9 x dr

Cassandra lives on an island in Massachusetts and wants to know its area Cassandra measures distances across the island at one mile intervals as shown below A B C D 465 6 Use Simpson s rule to approximate the area of the island Area square miles

F being a closed surface enclosing some simply connected volume V and for a vector field B x y z 2y 3x z use the divergence theorem Satz von Gau to calculate B da F

1 Test the convergence divergence of the following integrals i S ii iii iv 1 log 1 x 1 L sin 0 sin 1 x dx log x dx 0 x 2 cot rdr dx

2 T 4 x X So sin x t o t dt 1 Solve the integral equation

Exereises 25 and 26 Sketch the enclosed region and use the Shell Method to calculate the volume of rotation about the X asis 25 26 x y 4 y x 0 x y 4 y x y 2

A certain function f is continuous and satisfies f 3 0 The graph of f is shown below Find the following quantities f 0 f 7 Click on the graph for a larger version S f x dx Suggestion Sketch an accurate graph of f to support your reasoning

Given the piecewise defined h x e 2 for 3 x 0 ez 4 for 0 x 2 for all other 101 evaluate the integral Sh z dz function

exp 5t4 sin 2t cos 5t sin 11t 64 16 d

Find A the area of the region under the curve y 2x 2 and above the x axis for 3 x 8 Answer A

Find A the area of the region under the curve y 7 3x2 and above the x axis

The figure below to the left is a graph of f x and below to the right is g x 1 5 0 5 0 5 0 51 0 5 a What is the average value of f x on 0 x 2 avg value b What is the average value of g x on 0 x 2 avg value f x c What is the average value of f x g x on 0 x 2 avg value 0 5 2 0 5 0 5 5 g x

Evaluate the indefinit integral f x 6 5 3x dx

b Find the volume of the solid in the first octant that is bounded by the graphs of z 1 x y 3x and y 4

valuate 1 the integrals Cx et dx

Let f x 7x 3 if x 8 7x b if x 8 If f x is a function which is continuous everywhere then we must hav b

Differentiate the function y 2x logg x

7 Let p x be the sum of the first 2 non zero terms of the Taylor series for sin x centered at x 0 The maximum value of sin x p x for 1 2 x 1 7 is ANS 0 110 Show Transcribed Text

Question 28 9 2 40 BE Part 1 of 2 permutations to solve m a cooler with 9 different kinds of soda 5 are selected for 5 people In how many ways o stitute values into the formula for Pr points O Points 0 of 1 De an integer or a decimal

4 In x a Use l Hopital s rule to find lim x 2x In x x 2x X b Use l Hopital s rule to find lim

Find f given that f x 12x 6x 4 f 0 4 and 1 1 Ma

4 Find A f 4 1 F D C f x 2 find x X 3 E Symmetric to x axis y axis or origin G Increasing B f 2 1 x intercept s D Even or Odd or Neither R H Decreasing J y intercept s 3 2 1 3 2 1 K local max min absolute max min 4 1