Definite Integrals Questions and Answers

29 Solve for the unknown part of the triangle if it exists If b 125 c 178 and C 113 then what does a A 150 B 122 C 61 D 87

The density function for the number of times the riders scream on a roller coaster is given by 1 cos 4x if 0 x 2 otherwise 0 Find the mean number of screams over the course of the ride

48 Given that A 3 4j and B 51 12j find a unit vector parallel to A 1 A B 3 5 4v 5 D 3 4 3 4

49 Given that A 3 4J and B 51 12 find 3A 2B A 19 i 12 j B 19 i 28 C 21 i 24 j D 21 i 48 j

33 Change each number to algebraic form 3 cos 90 i sin 90 A 3i B 3 3i C 1 3i D 3i

Listen Use the cylindrical shells method to find the volume of solid obtained by rotating the region bounded by the given curves about the line x 1 y x y 0 x 1 x 4 255 2 510T 01020T 255T 510T 2 2T

T x sin x y dxdy Show Transcribed Text Evaluate the following integrals

Consider the definite integral a Approximate the definite integral with the Trapezoidal Rule and n 4 help numbers 19a 19x dx b Approximate the definite integral with Simpson s Rule and n 4 c Find the exact value of the integral

Solve the initial value problem 2 x y 3y 0 Assuming that the solution is of the form y x ao a1 a2 a3 a4 a5 a6 a7 help numbers help numbers help numbers help numbers help numbers help numbers help numbers help numbers k 0 y 0 0 akxk y 0 2 determine the following coefficients

Evaluate the definite integral 3 L 4x 1 4x 1 3x dx

5 f x 1 2x 1 dx 24 2x 1x 1 dx 23 2 12 2 1x 1 x2 x 2x2 2 2 31 32 3 29 b N 2 1 2 102

1 Estimate the area under the curve f x 2x 3 from a L AX b a7 3 0 20 b 46 FG flos f f l s f 2 f z 5 L6 3 4 5 6 7 8 2 2 16 S b R6 R6 F 5 f 1 f 1 5 f 2 f 2 5 f 3 A 4 5 6 7 8 9 39 19 3 26 c Find the exact area using Geometry 0 3 by finding L NI 0 0 5 1 1 5 2 2 5 3

Flu 5 31 2 de 3 amput F ix b Show F is F is 1 1 c Compute F 0

F x Fla St JEST DE 2 F is explain why F ompute F 0 00 x 00 is invertible

Suppose F t has the derivative f t shown below and F 0 2 Find values for F 3 and F 7 N 2 1 1 2 3 F 3 F 7 2 3 f 10 ba N

A candy of the type from the chart below is selected at random form each of two newly opened bags What is the probability that the two candies have the given color s Write your answers an decimal Color Proportion Brown 0 3 both are orange 01 Red Yellow 0 2 0 2 one is red and the other is green neither is yellow 64 04 Green 0 2 Orange 0 1

36 Suppose that f x y 15 000xe x is the population den sity of a species of small animals Estimate the population in the region bounded by y x y 0 and x 1

2 Step 2 of 2 Find two points on the graph of this function other than the origin that fit within the given 10 10 by 10 10 grid Express each coordinate as an integer or simplified fraction or round to two decimal places as necessary Answer Enter the coordinates to plot points on the graph k x A 5 Keyp Keyboard Shor X

Determine the volume of the frustum of a pyramid the pointed top cut off with a square base of side b 17 square top of side a 5 and frustum height h 19 Hint We proved in our notes p 6 that the volume of pyramid with a square base is V 1 h Use similar triangle to determine the overall height of the pyramid This is one way to do the problem without integration The integration formula is 2 ay by b a dy V Sh a 2257 none of these 2725 2572 2527

Verify that the Fundamental Theorem for line integrals can be used to evaluate the following line integral and then evaluate the line integral using this theorem Sv ex cos y dr where C is the line from 0 0 to In 3 2rt C Select the correct choice below and fill in the answer box to complete your choice as needed A The Fundamental Theorem for line integrals can be used to evaluate the line integral because the function is conservative on its domain and has a potential function p x y Type an exact answer OB The function is not conservative on its domain and therefore the Fundamental Theorem for line integrals cannot be used to evaluate the line integral

Evaluate the line integral v dr for the following function and oriented curve C a using a parametric description of C and evaluating the integral directly and b using the Fundar C Theorem for line integrals p x y 3x 5y C r t 2 t t for 0 t 2 Using either method Vop dr C Type an exact answer

For the following vector field compute a the circulation on and b the outward flux across the boundary of the given region Assume the boundary curve is oriented F x y R is the half annulus r 0 1srs3 0 0sm a The circulation on the boundary is Type an exact answer using as needed

2 me area enclosed by y x and y revolved about the line x 1

1 Evaluate 2xy dA where R is the region bounded within the stright lines connecting vertices 0 0 1 2 and 0 3

Evaluate y dr a dy along the following paths from 0 0 to 2 4 a the arc of the parabola y r b the horizontal interval from 0 0 to 2 0 followed by the ver tical interval from 2 0 to 2 4 c the vertical interval from 0 0 to 0 4 followed by the horizontal interval from 0 4 to 2 4 Solution a y dx x dy 1 x t y b dx dt dy 2t dt 72 y dx x dy x t y 0 L y dx 2 dy x y dx a dy 1 dx x dy 1 de dy dt 71 dx dt dy 0 dt 0 1 1 d 1 2 0 dt t 2t dt 72 5 0 t 2 0 t 0 dt 1 0 y dx x dy 16 t 0 dt 0 0 ost 4 Podt 2 dt 16

Find the surface area of revolution about the x axis of y 5x 6 over the interval 0 x

The arclength of y 3x 2 for 0 x 1 is

The length of the curve x et cos t y et sin t for 0 t 5 is

2 The arclength of y 3x 1 from x 0 to x 3 is given by OL OL OL Jo 1 36xdx 0 3 OL 3 fo f 6 da OL 3 1 6x da 0 Using the substitution u 6x to re write the integral in terms of u yields 18 1 OL VI 3u du 0 OL OL 3 6 1 1 36x2dx 6 1 3u du 18 u du 0 18 1 62 6 18 1 16 00 1 udu 1 6udu The value of this integral is

8 Find the area of the region enclosed by one petal of the curve r 2 sin 30 9 Find the area of the region enclosed by the curve r 3 2 cos 0

Consider the graph of U S House and Senate Reelection Rates 1964 2018 US House and Senate Roelection Rales 1964 2018 DE LAKE a Sanda kak www www Which of the following statements does the data in the graph support Gender is a better predictor of success in a congressional election than incumbency status A candidate s status as an incumbent is a weak predictor of success in congressional elections O An incumbent representat ve is slightly more likely to be reelected than an cumbent senator The incumbency advantage is only strong in the

5 points Evaluate the iterated integral f 1 S y x dyda Show all steps Do not reverse the order of integration

5 points Sketch the region of integration R and evaluate the integral over R in polar coordinates If you do not transform to polar coordinates to evaluate the will receive no credit Show all steps R r 0 2 r 4 0 0

Find all critical points of f a y Show all steps Express your final answer s as ordered pairs

The graph of y 6x 12 X 7 has a vertical asymptote What is it

Evaluate the following integral fi fidzdady

4 Solve on a calculator but show work be careful this one is problematic cot 5

9 Find the integral Scot x dx 1 x Ln x a C S dx

7 Suppose that R and h are changing that R is increasing at a rate of 2 m sec and h is decreasing at 4 m sec find rate of change in volume when R 10 m and h 20 m

Use the Gauss Jordan Elimination method to solve Change the system of equations to a matrix Get a 1 in row 1 column 1 Get a 0 in row 2 column 1 Get a 1 in row 2 column 2 2x 3y 9 Show your step

Click here to watch the video a Identity an equation that could be used to find the marked angles in each diagram L X 3x 15 2x 5 2x 5 3x 15 2x 5 3x 15 M L M 3x 15 2x 5 3x 15 90 L M 2x 5 2x 5 L 3x 15 M 2x 5 3x 150 M L M 2x 5 3x 15

Find the total area between the x axis and the curve f x ex 1 between x 5 and x 2 O e e 5 7 units O e e 5 units e 5 e 1 units e 5 e 1 units

Find the flux of the following vector field across the given surface with the specified orientation Use either an explicit or a parametric description of the surface F y x 1 across the cylinder y 4x for 0 x 2 0 z 3 normal vectors point in the general direction of the positive y axis The flux is Simplify your answer

Find the following indefinite integral Bescxcotx 25 Jdx 6cscxcc X O 6cscx O 6cotx O 6cotx 25 3x 25 3x3 25 3x3 C C C

Find the exact area between the graphs of x y and x 8 y 32 0 3 units O 16 units2 O 64 3 units 22 unito

Shown below is the graph of the flow rate f t of a pipe in gallons per minute Here t the number of minutes Using this information find the total number of gallons of water that flowed through this pipe in 7 minutes Velocity Gallons Minute 1 0 1 o frat IF 1 a of f 1 dt 1 2 f t dt 24 gallons f t dt 25 gallons So f t dt which gives 3 f t dt 26 gallons Time In Minutes 4 5 6

6 5 4 O x 2 3 2 Ox 3 x 0 1 10 x 3 x 0 x 2 0 1 2 1 2 Of x is differentiable for all x values 3 ark this question For the graph shown below find all values of x for which f x is not differentiable 4 5 6

ind the specified nth term in t 10z 7 n 5

Evaluate x 31 6 dx using the following graph 3 os 0 1x 31 6 dx 21 5 M S 1 31 x 31 6 dx 42 5 31 6 dx 17 5

Find the average value and the value s of c guaranteed by the mean value theorem for integrals if f x x 3 over the interval 0 3 O O O O favg c 3 and c 3 6 farg c 15 and c 15 C 18 favg 18 c 15 farg 6 3