Definite Integrals Questions and Answers

The following limit of right Riemann sums can be expressed as a definite integral n 7 7i 2 5 3 7 7 2 1 n n i 1 lim n In the integral b f x f a da

Consider the graph of f n 1 2 3 4 5 6 7 1 fl S f f x dx The shaded region over 0 2 has an area of 9 and following definite integrals f x dx f z dz 9 Q 0 1 pt 30 f x da 171 Use this information to evaluate th

Let If Use the given definite integrals above to evaluate each definite integral below 7 3f v dv 7 f v 6 du f v dv f v dv f v dv 108 f v dv 108

Suppose f x 7 10 a The rectangles in the graph on the left illustrate a left endpoint Riemann sum for f x on the interval 4 6 The value of this left endpoint Riemann sum is and it is an of the region enclosed by y f x the z axis and the vertical lines 4 and x 6 the area b The rectangles in the graph on the right illustrate a right endpoint Riemann sum for f x on the interval 4 6 The value of this right endpoint Riemann sum is and it is an area of the region enclosed by y y the c axis and the vertical lines and v the Left endpoint Riemann sum Right endpoint Riemann sum

Evaluate the integral 14 A dt N B C G 13 D

Find the partial fraction decomposition 2x 21x 21x 75x 96 2 x 8x 16 X 3 X

0 A 45 X y 2 9 3 B 27 9 3 C 36 D 18

dy 2 dt y t y 0 1 Aproximate yhe solution of IVP at t 1 by eu mothod b 05

206 986 zimsb nA CA TC shoitanut sint b Find Sex sec X fanx dx tosini stinitabal Pa Lidel 0 Hint Table 4 9 PP325 see the title page

5 cos x sin x dx

Calculate the following integrals 4x 5 2x dx 4x 5 2x dx mencerit sulky nes srltyd in A B C f 5 cos x sin x dx 4 x e 6x 2 x 1 2 dx

10 sing the Fundamental Theorem of Calculus Part 2 Interpret each result geometrically Evaluating definite integrals Evaluate the following definite 1 t 1 6x dx 1 16 So 60x Ass b eg 2T 530 3 sin x dx

Definite integrals from graphs Figure shows the graph of a tion f with the areas of the regions bounded by its graph and the x axis given Find the values of the following definite integrals x dx b 1 x dx a C f x f x dx d y A a f x dx 12 Figure y f x b 10 C d

3 and Properties of integrals Assume that fof x dx of x dx 10 Evaluate the following integrals if possible f2f x dx b f f x dx e So f x d c ff x dx d f6f x dx e

43 44 Evaluate the integral and interpret it as the area of a region Sketch the region 43 sin x cos 2x dx 44 13 2 dx

Solve the initial value problem dy da 1 8 T y S

Simplify the following trigonometric expression tan x cos x csc x 01 cot x O sin x sin x

1 8 Evaluate the integral by making the given substitution 1 cos 2x dx u 2x

1 Evaluate the integral 4x y z dxdydz

The distribution of IQ scores is often modeled by the normal distribution with mean 100 and standard deviation 15 a Write a formula for the density distribution of IQ scores 1 15 2 T p x x 100 450 07 e b Estimate the fraction of the population with IQ between 114 and 119 NOTE Round your answers to one decimal place

I Find dz where C is the line segment from 2 i to i af z fon

Find the median of the density function given by p t Round your answer to one decimal place The median is Toxtbook and Media 0 04 0 0008t for 0 t 50 using the Fundamental Theorem of Calculus

Evaluate the following integral 2 sipx IIT siny dz dx dy 000 x 2 sipx siny dz dx dy 00 0 Simplify your answer Type an exact answer using as needed Use integers or fractions for any nu the expression

Evaluate the following integral 341 N LEX x y dz dx dy dz dx dy Simplify your answer www

oblem 6 Evaluate the following sum n 1 2i A n 4n B n 4n 10 C n 5m 15n 5 D n 4n 1 E n 5n 13 F n 5n 18n 2 G n 5n 15n 1 H n 4n 4

6 7 2965 14 3 2 14 1 24 3 4 5 6 What is the period for the graph show above What is the vertical stretch for the graph shown above Write an equation for the function graphed above Use y for the output

Consider the function f x 11 x Find the Taylor polynomial of degree 1 for f about x 14 y Again note that this is just the linear approximation of fat x a

Evaluate the following indefinite integrals 3 1 dx 3 b 5x dx 4 4x

While taking a walk along the road where you live you accidentally drop your glove but you don t know where The probability density p x for having dropped the glove x kilometers from home along the road is p x 2e 2 for x 0 a What is the probability that you dropped it within 1 kilometer of home NOTE Enter the exact answer or round to three d The probability is b At what distance y from home is the probability that you dropped it within y km of home equal to 0 95 NOTE Round your answer to three decimal places y nal places km

The probability of a transistor failing between t a months and t Your answer is correct C a If the probability of failure within the first six months is 10 what is c Round your answer to four decimal places 0 0176 of c b months is given by c et dt for some constant c

57 The area of the region that lies to the right of the y axis and to the left of the parabola x 2y y the shaded region in the figure is given by the integral 2y y dy Turn your head clockwise and think of the region as lying below the curve x 2y y2 from y 0 to y 2 Find the area of the region YA 2 0 x 2y y

44 ST 4 3ex 0 3e 4 sec x tan x dx

Evaluate the integral above by taking the derivative of each answer choice to determine which function is the anti derivative of 5 Select the correct answer below O 5 In 5 C In s C 5 Ox 5 C 5x 1 C 5 dx O 5x 1

25 54 Evaluate the integral 3 25 x 2x 4 dx 27 1 3t dt 29 x dx 31 33 0 1 1 dt cos e de 2 26 28 30 32 S 1 0 100 X dx 1 8v 16 fx 2 3 dx 3 1 L dz 2 3 5 34 Se dx

4 Imagine you are driving a car on a highway and your distances in meters from a certain point is given by the function s t 30t 2t where t is the time in seconds since you passed that point a Determine the time at which you pass the point for the second time b Calculate your average speed over the first 10 seconds after you pass the point for the first time c Find the limit of your average speed as t approaches 0 Interpret the meaning of this limit in the context of the problem Hint Average speed over an interval a b is given by s b s a b a

25 54 Evaluate the integral 25 x 2x 4 dx 27 31 1 31 dt 29 S x dx 26 fx100 dx 28 1 8v 16v du f 10 30 S 2 3 dx

In each part find a formula for the general term of the sequence starting with n 1 Enter the following information for an a an b an c an d 111 n 1 1 1 1 3 n 1 1 1 3 0 1 1 1 3 9 27 2 1 1 9 27 2 8 26 80 2 3 9 27 81 1 4 9 2 2 V v v 9

1 point Use the Integral Test to determine whether the infinite series is convergent n n 10 n 9 To perform the integral test one should calculate the improper integral dr Enter inf for inf for oo and DNE if the limit does not exist By the Integral Test the infinite series OA converges OB diverges n n 10 n 9

Find an arc length parametrization of the line y 72 12 Use symbolic notation and fractions where needed Give your answer in the form of comma separated list of 1 j components Fi 9 a help fractions

work No credit for digital work 1 6 points Assume f x dx 3 f f x dx 5 and f3 g x dx 9 compute 3 3

9 20 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function 9 g x t t dt 11 9 w sin 1 1 dt 13 F x 1 sect dt 10 g x f ln 1 1 dt t t 1 12 h u Hint 1 sec t dt f 1 sect dt dt

An artist wants to paint the interior of the shape shown below on the side of a building Meters y y 3 x Meters 9 X How much area in square meters will the artist need to paint Round your answer to two decimal place m

Solve the equation for x Use n as an integer constant cos r cos r 1 0 X X smaller value for n 0 larger value for n 0

Find the 9R9 of the finite region between the x axis and the Curve y x x 2 x 5

Current Attempt in Progress The concentration C in ng ml of a drug in the blood as a function of the time t in hours since the drug was administered is given by C 15te 0 21 The area under the concentration curve is a measure of the overall exposure of a person to the drug Find the total exposure provided by the drug between t 0 and t 5 Round your answer to two decimal places Exposure Modia ng ml hours Attempts 0 of 5 used Submit Answer

Example find the the area within r s100 T 203 3 20 27

Find the area under one arch of the cycloid x 2a t sin t y 6a 1 cos t The area is Type an exact answer using as needed

Find the global maximum value of f x 6ce 2 on the interval 6 5 The global maximum occurs at x 0 and the maximum value is y

Co US Secti Consider the equation below If an answer does not exist enter DNE f x x4 8x 7 a Find the interval on which f is increasing Enter your answer using interval notation 2 Find the interval on which f is decreasing Enter your answer using interval notation b Find the local minimum and maximum values of f local minimum value 9 local maximum value 7 Par c Find the inflection points 2 V 3 2 x y 3 3 x smaller x value BB x larger x value X Find the interval on which fis concave up Enter your answer using interval notation 2 2 U 00 3 Find the interval on which fis concave down Enter your answer using interval notation 2

41 46 Evaluate the integral by interpreting it in terms of areas 42 f 2x 1 dx 44 2x 1 dx 46 2x 16 x dx 41 10 5x dx 2 43 4 15 1 9 x dx 3 x dx