Definite Integrals Questions and Answers

For the function f whose graph is shown list the following quantities in increasing order from smallest to largest ya 2 0 EN A Rx dx D F X f x dx OC D A E B OB E A D C OE E C D B OE B D C A OB D CSASE 5 B f x dx E f 1 c fix dx

15 lim 84x 3 n 1 7 1 3 n 1 2 3 n 1 1 1 3 3 n I

3 Give the precise definition of a definite integral ff x dr in terms of limit of a Riemann sum then explain what does f f x dr represent

4 The region bounded by the prabola y x and the line y 4x in the first quadrant is revolved about the y axis to generate a solid Find the volume of the solid

Find the derivative of the function F x 1 e 2 o de dt F x X 4

Let F x Loc dt Find an equation of the tangent line to the curve y F x at the point with x coordinate 3

Find the derivative of the function g x y sin x Jou cos x In 7 5v dv

Evaluate the integral 192 x 3 X dx

Let g x f t dt where f is the function whose graph is shown g 8 g 12 g 24 y 16 12 8 4 4 8 f a Evaluate g 0 g 4 g 8 g 12 and g 24 g 0 g 4 4 8 12 16 20 24 28 t b on what interval is g increasing Enter your answer using interval notation c Where does g have a maximum value

3x 2 on 1 x 3 Find the arclength of y 3x 2

Let C be the curve y 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Surface area 2 f x Find the surface area of revolution about the x axis of R 3 3 1 1 5 7 x for 1 5 x 3 3 f x dx where

Evaluate the definite integral 3 1 x 2 3 dx

If f x 3 cos x Find the width of each subinterval T 8 x2 3 Find the left endpoints of the subintervals where x x x6 x1 0 4 in x6 7 8 KA KIN 3 3x 8 31 4 units 5x 2 0sxs evaluate the left Riemann sum with n 6 taking the sample points to be left endpoints Evaluate the Riemann sum Round your answer to six decimal places 3 0995 What does the Riemann sum represent The Riemann sum represents the net area of the rectangles with respect to the x axis Illustrate with a diagram y 3x 3 3x 5A 8 3x 4 1 00 214 3x 8

c Consider the following theorem If f is integrable on a b then Use this to evaluate the integral 5 33 y 9r 8 7 6 5 4 3 2 1 x d Interpret the integral in part c as a difference of areas O The integral represents the area below the x axis minus the area above the x axis O The integral represents the area above the x axis minus the area below the x axis Illustrate with a diagram 1 2 0 34 2 3 f x dx lim f x Ax where ax 318 i 1 5 5r 4 3 2 1 1 2 3 b a and x a iAx n 4 6 st 2 4 5 10r 9 8 7 6 5 4 3 2 0 14 1 2 3 4 5 9r 8 7 6 5 4 3 2 2

Tutorial Exercise Use the Midpoint Rule with the given value of n to approximate the integral x 6 dx n 4 Step 1 We must calculate M4 f x ax F x F X f x3 f x4 ax where X X X3 X represent the midpoints of four equal sub intervals of 2 10 Ax 1 1 Since we wish to estimate the area over the interval 2 10 using 4 rectangles of equal widths then each rectangle will have width Ax No Response Step 2 Using f x x 6 to find M 2 x 6 x 6 x 6 X3 Since X X X3 X4 represent the midpoints of the fo x X3 4 sub intervals of 2 10 then we must have the following

If w t is the rate of growth of a child in pounds per year what does 13 s frow t dt w t dt represent J5 O The child s weight at age 5 O The child s weight at age 13 O The change in the child s weight in pounds between the ages of 5 and 13 O The child s initial weight at birth O The change in the child s age in years between the ages of 5 and 13 W

If f x kx determine the value of k that makes f x a probability density function on 0 x 7 3 The value of k that makes f x a probability density function is 343 Type an integer or a simplified fraction

Evaluate the definite integral 3 4 csc t cot t dt

Evaluate the definite integral 3 6 7 sec y dy

The length of time in years until a particular radioactive particle decays is a random variable t with probability density function defined by f t 5e 5t for t in 0 00 Find a the mean of the distribution b the standard deviation of the distribution and c the probability that the random variable is between the mean and the 1 standard deviation above a The mean of the distribution is years Round to two decimal places as needed b The standard deviation of the distribution is Round to two decimal places as needed c Set up the integral necessary to find the probability that a random variable is between the mean and 1 standard deviation above the mean Sat years What is the probability that a random variable is between the mean and 1 standard deviation above the mean The probability is Round to four decimal places as needed

Find the derivative of the function dy dx y 2e5x 3 11

f x 2 cos x 0 x 3 evaluate the left Riemann sum with n 6 taking the sample points to be left endpoints Round your answer to six decimal places What does the Riemann sum represent The Riemann sum represents the net area of the rectangles with respect to the Select Illustrate with a diagram 3 pr 8 5 3x 4 0 2 4 4 3 8 5x 3 8 4 y 0 2 00 14 28 4 37 8 5

If Rx 3 cos x 0 x Find the width of each subinterval 2 Find the left endpoints of the subintervals where x x x6 x 4 x units x6 evaluate the left Riemann sum with n 6 taking the sample points to be left endpoints Evaluate the Riemann sum Round your answer to six decimal places What does the Riemann sum represent The Riemann sum represents the net area of the rectangles with respect to the Select V Illustrate with a diagram 3x MA CO 4 00 8 X 4 3A 8

Find the Riemann sum for f x 2x 1 6 x 4 with five equal subintervals taking the sample points to be right endpoints Explain with the aid of a diagram what the Riemann sum represents y Need Help 7 Read II 5 1 31 5 7 9 11 13 2 The Riemann sum represents the net area of the rectangles with respect to the Select y 7 5 3 9 11 13 X 6 O 4 2 y 7 5 3 3 9 11 13 X 6 11 ASM y

Evaluate the following 2 f v 4 dx V8 X

Find the derivative of the following function dy dx y 106x 1 11

Graph of a function y f x is shown Evaluate 2 S 4 f x dx Show your solution co y f x 2 1 3 2 0 1 2 3 Selected Answer None Given Response Feedback None Given 3 4 E

Evaluate the integral 5te4 i 8e 5t j 6 k dt 1 0 5te4 i 8e 5t j 6 k dt i j k Be 0 Type exact answers

Evaluate the integral T 3 3 3 2 sin t i 3 4 cos t j sec t k dt 2 sin t i 3 4 cos t j sec t k dt sec t k dt i Dj k 3 Type exact answers

Evaluate the integral 1 S 1 i 0 i 4 j 5t 2 k dt S i i 4 j 5t 2 k dt i 0 TYDO OYst

Evaluate the integral e dA The region R is shown below

6 2 Evaluate So Sex y 2 dydx Hint you may have to change the order of 2 Integration so it might help to draw the region of integration and rewrite the integral as dxdy

Match the equation with its graph y 2x 2 O no x DO DOTA

A 5 inch rod is placed along the z axis starting at x 2 The mass density of the rod 1 Find the mass of the rod 1 r is p x

Set up the integral of the surface area obtained by revolving the region x q bounded by the graph of y 3a 4x B D E 2a 27 2T a 2a 27 2a 277 J S a 24 27 2a X a 3a 3 16x X a 4 3a 3 16x x 4 x 4 3a X Ba 4 Ba x a 3 16x 36 3 x 16x a 3 16x dx dx dx dx between x a and x 2a about x axis 2023 3925870766

Solve the equation Find only the real solutions Enter your answers as a comma separated list 4x 5 x 1 2 3 5 x 0

27 Estimate the definite integral 1 z dr using eight rectangles and right endpoints

40 Given that f x sin t 5 dt find f r 2

Find the indefinite integral Note Solve by the simplest method not all require integration by parts Use C for the constant of integration 1 11 arctan x dx 11arctan x x ln 1 c 2

1 x 1 1 3 using 8 approximating rectangles and right endpoints Estimate the area under the graph of y over the interval

12 Find the volume of the solid generated by rotating the region bounded by y sin x and y 0 between 0 and about the x axis

13 Sketch the region between y 5x and y 3x Find the area of the region

9 Evaluate the definite integral T 6 e4s 0 4sinx COS xdx

Find the area under the curve y over the interval 1 2 X 2 8 7 6 5 4 3 2 N

7 Find the average value of the following function on the given interval f x x 1 x on 0 1

One number is seven more than six times another number If the sum of the numbers is 28 find the numbers Provide your answer below

One numberis eight more than three times another number If the sum of the numbers is 28 find the numbers Provide your answer below

Evaluate the definite integral Use a graphing utility to verify your result 12 22 16 2x x 25 dx

FICO ase 3 roo dr 2 the graph evaluate 2 2 Y F x 2 3

Estimate the definite integral 22 da using four rectangles and right endpoints 1 x Answer 7 28