Definite Integrals Questions and Answers

Consider the area between the graphs x 4y 28 and x 4 y This area can be computed in two different ways using integrals First of all it can be computed as a sum of two integrals with the following values a b C f x g x with the following values Alternatively this area can be computed as a single integral X f x B h y f x dx 9 x dx Sh 3 dy

Sketch the region enclosed by the curves given below Decide whether to integrate with respect to x or y Then find the area of th region Area y 4 y e x 1

Find c 0 such that the area of the region enclosed by the parabolas y x c and y c is 10

A company finds that the marginal profit in dollars per foot from drilling a well that is x feet deep is given by 6 P x 3 x Find the profit when a well 300 ft deep is drilled Set up the integral for the total profit for a well that is 300 feet deep P 300 dx 0 The total profit is Round to two decimal places as needed

3 3 5x 2 dx 2 Type 5x 2 dx an integer or a simplified fractio

Find the area under the graph of the function over the interval given f x ex 1 2 The area is Type an exact answer in terms of e

Find the area under the given curve over the indicated interval 2 y x 3x 3 2 5 The area is

226 Suppose that a particle moves along a straight line with acceleration defined by a t t 3 where 0 t 6 in meters per second Find the velocity and displacement at time t and the total distance traveled up to t 6 if v 0 3 and d 0 0

Find Ln Rn and their average for the definite integral below using the indicated value of n 7 5x x dx n 6 4 The left Riemann sum Ln is Do not round until the final answer Then round to three decimal places as needed The right Riemann sum R is Do not round until the final answer Then round to three decimal places as needed The average of the two sums is Type an integer or decimal rounded to three decimal places as needed

224 Suppose that a particle moves along a straight line with velocity defined by v t t 3t 18 where 0 t 6 in meters per second Find the displacement at time t and the total distance traveled up to t 6

Evaluate S 6 2x 1 2 3 X x 7 2x 1 20 dx dx In 7 x x 7 2 Type an exact answer

In the following exercises use the evaluation theorem to express the integral as a function F x fe 190 t dt

In the following exercises evaluate each definite integral using the Fundamental Theorem of Calculus Part 2 1 1 2 3a da 170 L 171 172 t 2 t 3 dt 1 9 4 1 dt 173 174 24 f 2 da x dx S 175 x 3x 5 dx 176 4 5 2 36 3 2 dt 177 x dx 1 22 da

Use the Midpoint Rule to approximate with n 7 4 5 2 5 x dx

A Estimate the area under the graph of f x 36 x from x 0 to x 6 using 6 approximating rectangles and right endpoints Estimate B Repeat part A using left endpoints Estimate C Repeat part A using midpoints Estimate

Evaluate the integral by interpreting it in terms of areas x dx

8 5 3 5 Lot f f x dx 6 f x dx 6 Find f x dx 3 5 3 5 and 6 6 f x 6 dx 8 5 f x da 3

Given that 4 f x 6 for 4 x 7 use property 8 of Section 5 2 of the Stewart Essential Calculus textbook to estimate the value of f x dz O f x dr

Speedometer readings for a motorcycle at 12 seconds intervals are given in the table t sec 0 12 24 36 48 60 v ft sec 26 22 26 27 30 20 a Estimate the distance in feet traveled by the motorcycle during the time interval from 0 to 60 seconds using the velocities a the beginning of the time intervals b Give another estimate using the velocities at the end of the time periods

Find the area of the region between y x 2 and y x 5 for 0 x 1 area

Evaluate the given indefinite integrals You may find it useful to expand the integrand before proceeding a 7x 7 dx b 3x 7 8x 6x dx C C

Evaluate the given indefinite integrals aftdt t dt a b C d 8t dt It at S e dt 10t5 dt 8t 10t5 10 dt C C C C C

1 2 Find the area of the shaded region 2 1 YA y 5x r2 y x 4 4 X

Question 3 of 10 1 point Zoom Meetings b Solve the right triangle for the unknown sides and A 29 1 b 19 1 Attempt 1 of Unlimited C

ege Homepage Question 2 of 10 1 point A Zoom Meetings Solve the right triangle for the unknown sides and ang a 9 b 9 9 b Attempt 1 of Unlimited

Also find the x dx using the Riemann Sum formula b f x dx lim x Ar 81x i 1

In the following exercises find the average value fave of f between a and b and find a point c where f c fave 110 f x x a 1 b 1 111 f x x5 a 1 b 1 112 f x 4x a 0 b 2

In the following exercises given that integrals 1 x x x dx 99 1 2 x 2 de 98 0 f 1 x dx 100 fo t xe xdx of 1 2 x dx 3 and x dx 1 compute the

Suppose f x 1 33 x 6x 8 x If f 8 185 what is f 1 equal to Enter a

3 Suppose f s 25 1 4 and f 1 S S 25 2 What is the constant term C in f s equal to Enter a number only

Question 8 Instructor created question suming that b is positive solve the following equation for b 30 3x 10 dx 16 Round your final answer to 4 decimal places Give the positive answer only

Let b t represent the population of bunnies on Greentree Island t weeks after the population was first re Assume that b 0 184 b 20 1284 b 40 1347 20 10 b t dt 307 b t dt 880 B A What was the population on the island exactly 10 weeks after the population was first recorded 10 30 b t dt 40 20 C What was the population on the island exactly 30 weeks after the population was first recorded D b t dt 30

In the following exercises evaluate the integral using area formulas 3 0 76 3 x dx

The duration of a scuba dive in minutes is given by T v D 33v D 33 where v is the volume of air in the tank at sea level pressure and D is the depth of the dive in feet Evaluate T 86 32 and T 79 38 T 86 32 minutes Round to the nearest minute as poodod

Evaluate the definite integral 0 Sy4 5y5 6 dy 1 0 y sy 6 dy Simplify your answer Type an exac 2

Evaluate the definite integral 11 50 4 5u 1 4 1 u 5 du 1 u 5 Type an exact answer in terms of e du

Evaluate the definite integral 9 9 4 40y y 15 y dy 40y y 15 ydy

Evaluate the integral 5 2x 6 6 dx 3 5 2x 6kdy 12x

Find the exact value of the integral 8 64 0 64 x dx 1

Approximate the area under the gra f x x2 8 from x 2 t a Use left endpoints b Use right endpoints c Average the answers in parts d Use midpoints

Integrate 3 S In 2x dx ww 3 S In 2 1 Do not round until the final answer Then round to four decimal places as nee In 2x dx

For each of the described curves decide if the curve would be more easily given by a polar equation or a Cartesian equation write an equation for the curve a a circle with radius 3 and center 1 3 b a circle centered at the origin with radius 1

Find the equation of the parabola that satisfies the given conditions Focus is on the x axis passes through 3 2 and 2 2

Find the arc length of the polar curve r a 1 cose

Find the volume of the solid bounded below by the circular cone z 1 y and above by the sphere x y 2 2 25z

Find the exact length of the curve x et e t y 5 2t 7 2537 0t

EXAMPLE 6 A particle moves along a line so that its velocity at time t is v t t t 6 measured in meters per second a Find the displacement of the particle during 1 t 6 b Find the distance traveled during this time period SOLUTION a By this equation the displacement is s 6 s 1 L v foce Live v t dt t t 6 dt v t dt coco This means that the particle moved approximately 24 17 meters to the right b Note that v t t t 6 t 3 t 2 and so v t s 0 on the interval 1 3 and v t 0 on 3 6 Thus from this equation the distance traveled is 1 1 2 620 v t dt Lovce v t dt 2 x 6 a 2 t dt t t 6 dt

Find the exact length of the curve described by the parametric equations x 2 3t y 3 2t 0 ts1 2 3 2 ln 1 2 X Enhanced Feedback Please try again keeping in mind that the arc length formula for para CB

Find the distance traveled by a particle with position x y as t varies in the given time interv x 3 sin t y 3 cos t Osts 2m Compare with the length L of the curve

A z So If 0 A A 31 1 3 3 1 t 11 dt then