Definite Integrals Questions and Answers

Find the line integral of f x y ye along the curve r t 4t i 3t j 1sts 1 he integral of f is Type an exact answer

Find the line integral along the given path C fix x C y dx where C x t y 7t 2 for 0 st 4 x y dx

Evaluate x y ds where C is the straight line segment x 2t y 8 2t z 0 from 0 8 0 to 8 0 0 f x y ds 0

Find the area of the region bounded by y 1 1 x2 y 2 x 1 and the y axis Find the area of the region bounded by x 6y2 4y x 3y 2

Use a calculator to solve the following equation for 0 on the interval csc 0 2 Find all the correct answers Report your answer as a decimal and round to three decimal places Provide your answer below

et C be the positively oriented square with vertices 0 0 1 0 1 1 0 1 Use Green s Theorem to evaluate the line integral J10y z dz 10x ydy

Given the side length b 9 and the angle A 73 on the triangle below find the lengths of a and c and the measure of angle B Do not round during your calculations but round your final answers to one decimal place B Provide your answer below a 73 A 9 C

Find the area of the shaded region x y 10 y 75 4 15 2 30 20 x 5y y 10 10 5 6 y X

Ow dy X and ow dy 2 2 at the point w x y z 12 2 1 2 if w 2x y 2yz 2z and x y 2 9

Find the area of the shaded region 1 y 8 00 6 4 2 1 1 y 5x x y 2x 2 3 3 6 4 5 LE 6 X

Use a substitution to find You must show your work to receive full credit In a Edit Format Table dx

Use a substitution to evaluate the definite integral 0 1 In 3 01 In 2 Cos 11 sin dr

3 15pts Evaluate cos B where cos 3 4 with in 2th Quadrant and sin 5 7 with B in Quadrant II 4 16 pts Find an equivalent expression for cot

Use the figure to evaluate the definite integrals g 1 dt and 2 1 2 8 1 a 1 g t dt y g t 2 3 4 5 8 Express numbers in exact form Use symbolic notation and fractions where needed g t dt

Sketch the region of integration associated with the double integral S S f x y 1 X Choose the correct sketch of the described region OA f x y dydx N 00 M OB 2 15 OC 5 Q 27 OD 5

Evaluate the integral x x 1 x 1 x 2 dx

Evaluate the integral 2 6 2x 3x 3 dx 1

a b c d dx X 7 S Since the function y Since the function y Since the function y Since the integral dx x 9 Since the integral Since the function y Since the function y Since the integral 8 S Since the function y Since the integral tan x dx 8 5 Since the integral Since the integral ex dx X Since the integral 10 Since the integral 1 Since the function y 1 Since the function y Since the function y 1 Since the function y tan x has an infinite discontinuity at x 0 the integral is a Type 2 improper integral Since the function y tan x has an infinite discontinuity at x the integral is a Type 2 improper integral 2 Since the function y tan x has an infinite discontinuity at x 1 the integral is a Type 2 improper integral 1 X 7 00 dx 1 has an infinite discontinuity at x 1 the integral is a Type 2 improper integral x 7 1 has an infinite discontinuity at x 7 the integral is a Type 2 improper integral X 7 X 00 dx has an infinite interval of integration it is a Type 1 improper integral x 9 has an infinite interval of integration it is a Type 2 improper integral x 9 dx 1 dx X 7 1 1 has an infinite discontinuity at x 9 the integral is a Type 1 improper integral x 9 1 has an infinite discontinuity at x 8 the integral is a Type 1 improper integral x 9 1 has an infinite discontinuity at x 7 the integral is a Type 1 improper integral x2 has an infinite discontinuity at x 9 the integral is a Type 2 improper integral 9 has an infinite interval of integration it is a Type 1 improper integral has an infinite interval of integration it is a Type 2 improper integral tan x dx has an infinite interval of integration it is a Type 1 improper integral tan x dx has an infinite interval of integration it is a Type 2 improper integral X X ex has an infinite discontinuity at x 1 the integral is a Type 1 improper integral X has an infinite discontinuity at x 0 the integral is a Type 1 improper integral ex has an infinite discontinuity at x 0 the integral is a Type 2 improper integral X X ex dx has an infinite interval of integration it is a Type 1 improper integral ex dx has an infinite interval of integration it is a Type 2 improper integral

c tan x dx 11 Since the function y tan x has an infinite discontinuity at x 0 the integral is a Type 2 improper integral Since the function y tan zx has an infinite discontinuity at x the integral is a Type 2 improper integral Since the function y tan x has an infinite discontinuity at x 1 the integral is a Type 2 improper integral O Since the integral tan x dx has an infinite interval of integration it is a Type 1 improper integral tan x dx has an infinite interval of integration it is a Type 2 improper integral Since the integral

The population densities of a rectangular county are shown in the figure to the right a Use the fact that population population density x area to estimate the population of the county b Explain how the calculation of part a is related to Riemann sums integrals a The population of the county is 3 2 3 2 Population densities have people per square m 350 450 250 1 2 100 450 250 3 2 100 250 100

To test the series is the case with all series of this type According to the P test 8 k 1 1 k3 1 for convergence you can use the P test You could also us k3 1 k3 converges the P test does not apply to diverges k3 Now compute s3 the partial sum consisting of the first 3 terms of 8

4 A population P of bacteria is growing at the rate of dp dt 3000 1 25t b What is the population after 3 days where t is the time in days When t the population is 1000 a Find a model for the population Hint integrate dP dt and find the particular solution

2 Locate any relative extrema and points of inflection y xlnx

Use the transformation u 4x 2y v x 2y to evaluate the given integral for the region R bounded by the lines y 2x 2 y 2x 3 y 2x and y x x 1 2x 5xy 2y dx dy R 2x 5xy 2y dx dy R Type an integer or a simplified fraction

T Find the volume of the portion of the solid sphere p 5 that lies between the cones 4 The volume is Type an exact answer using and radicals as needed 3 and 4

T 2T Find the volume of the portion of the solid sphere p 2 that lies between the cones 3 and 3 The volume is Type an exact answer using and radicals as needed

0 2 4 3 0 2 SS S p sin 24 do d0 dp 2 0 2x 4

Ketch the graph described by the following cylindrical coordinates in three dimensional space 1 7 Choose the correct answer below A OE OB COLLE O C OG 246 OD 14 OH

Using the accompanying Weddings data find the mean median and standard deviation of the wedding costs What would you tell a newly engaged couple about what cost to expect Consider the effect of possible outliers in the data Click the icon to view the Weddings data For the given data set the mean of the wedding costs is 25772 The median of the wedding costs is 24400 The standard deviation of the wedding costs is 13205 Type integers or decimals rounded to two decimal places as needed mild outlier s for the costs Consider the effect of possible outliers in the data There is are Remove these outliers and recalculate the mean median and standard deviation Recalculating these statistics the mean of the wedding costs is The median the wedding costs is The standard deviation of the wedding costs is Type integers or decimals rounded to two decimal places a needed 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 53000 46400 41800 33600 30100 29800 29300 27200 27000 24800 24600 24400 21400 20300 20300 20300 18600 18400 18100 15400 14400 13000 6000

Use Lagrange Multipliers to find the maximum and minimum values of f x y z x 3y ubject to the constraint x 4y 2 36

ts 3 Consider the curve C defined by the polar equation r 2 sin 30 1 Set up the area A of the region bounded by the curve C depicted below DO NOT EVALUATE THE DEFINITE INTEGRAL

5 Find the coordinate matrix x given the coordinate matrix x B 1 1 9 9 1 1 1 9 1 B 3 0 3 3 3 0 0 3 3 x B 5 4

Use a reference angle to write cos Provide your answer below 43 36 in terms of the cosine of a positive acute angle in radians

Evaluate the volume of the following obje 2 3y 2z 6 3x 2 3 Ty 2 2x

s 16 Round your answer to two decimal places 9 Compute the following using a calculator cos

Evaluate the integral 6 0 2 SS 0 2x 4 Sp 3 p sin 20 d d0 dp 6 0 2 SS S 0 2 4 Type an exact answer using as needed p sin 2 dbd0 dp

Find the volume of the portion of the solid sphere p 2 that lies between the T 5 6 cones 6 and The volume is Type an exact answer using and radicals as needed

Use a substitution to find 0 7 0 It 150 13 S cos 2 de x cour dix

Use the integration by parts formula to find freda Show your work and final answer in the provided area using HTML Editor s Insert Math Equation function Edit Format Table 12pt Paragraph BIUA 2 T

Find the arclength of y 2x 2 on 0 x 3

Find the surface area of revolution about the x axis of y 2 sin 5x over the interval 0 x 5

The force on a particle is described by 9x 6 at a point x along the x axis Find the work done in moving the particle from the origin to x 9

If the terminal side of angle 0 goes through the point 3 3 2 3 13 13 Provide your answer below on the unit circle then what is sin 0

The graph of f is shown in the figure to the right Let A x f t dt and 2 X F x f t dt be two area functions for f Evaluate the following area functions 4 a A 2 b F 8 c A 4 d F 4 e A 8 a A 2 0 Simplify your answer b F 8 9 Simplify your answer Area 9 Area 16 Area 91

An arch is 615 ft high and has a 560 ft base It can be modeled by the parabola 27 X y 615 1 0 Find the average height of the arch above the ground 280 700 600 700 400 300 200 100 400 200 0 200 400

Find the volume generated when the region between the curves is rotated around the given axi Answer exactly x y and y x rotated around the line y 8

The following integral is a O b OC x cos x 2 ex dx Convergent using D C T by comparison to Divergent using D C T by comparison to 580 201 dx 8 S 1 dx Convergent by using D C T by comparison to 3 1 dx

Evaluate F dr where is the line segment from 7 4 1 to 1 41 7 and 2 F yz cos xz 2x i sin xz 1 j xy cos xz 2z k

Use the Table of Integrals to evaluate the integral Remember to use absolute values where appropriate 11 1 12 15 dy

Use the Table of Integrals to evaluate the integral 1 dx 9x 4 C