Definite Integrals Questions and Answers

6 Using rules to combine known integral values 4 5 7 5 4 5 Suppose f a da 10 f x dx 8 f x dx 10 9 6 f x dx 7 5 7 5 10f x 8 dx 9

H N 2 5 pts 6 3 148 Sketch the region bounded by x y and y x Determine the volume when the region is rotated about the line y 2 2 a N only x Sinhly

1 Evaluating definite integrals from graphical information Use the following figure which shows a graph of f x to find each of the indicated integrals Click on the graph for a larger version Note that the first area with vertical red shading is 18 and the second with oblique black shading is 6 A S f x dx B ff x dx C ff x dx D ff x dx

6 5 pts 6 2 86 Sketch the region bounded by the curves y x 0 ans x 3 Then determine the volume when the region is rotated around the y axis 6 4 2 6 4 2 2 ly b 2

4 5 pts 6 1 17 Sketch the region bounded by y x and y x 2x over 1 1 Determine the entire area of the region 4 2 6 4 2 2 4 2 16 x K

Evaluate the definite integral 2 6 2x 4 x dx

ate the definite integral 3 8 8x 6 dx

Evaluate the definite integral S 4 sin x dx 0

A 25 year old woman burns 350t cal hr while walking on her treadmill How many calories are burned after walking for 4 hours calories burned

6 Find the area enclosed by 1 2 cos 40

The temperature at a point x y z of a solid E bounded by the coordinate planes and the plane x y z 1 is T x y z 9xy 8z 20 degrees Celsius Find the average temperature over the solid Answer to 3 decimal places Average Value of a function using 3 variables N y X

Let the region R be the area enclosed by the function f x ln x and 1 Find the volume of the when the region Ris g x x solid generated 6 You may use a calculator and round to the nearest thousandth revolved about the line y A 2 i I 1 14 12 10 8 10 6 4 2 y I 1 3 I 4 I 5 16 1 7 1 I i 1 8 9 10 X

The shape and density of a thin shell are indicated below Find the coordinates of the center of mass Shell nose of the paraboloid x y 2z cut by the plane z 5 Density 8 1 O A 2 y 1 OB 115 36 0 0 27 23 OC 0 0 27 OD 0 0 200 9 www

8 Below is the graph of a function f x with domain 2 6 4 07 2 2 2 1 15 2 37 0 85 a 2 points Decide whether the quantity justify your answer 0 85 2 72 3 4 4 4 0 85 3 4 5 656 f x dx is positive negative or zero Brieft b 2 points Decide whether the quantity f a da is positive negative or zero Briefly justify your answer c 2 points Decide whether the quantity f x dx is positive negative or zero Briefly justify your answer 5 65 d 2 points Compute f x dr Briefly explain why your answer is correct

Suppose that profits on the sale of swimming suits in a department store are given approximately by the following function P t where P t is profit in hundreds of dollars for a week of sales t weeks after January 1 where 0 st 104 t P t 6 6cos 26 A Use definite integrals to approximate the total profit earned during the 2 year period The total profit earned during the 2 year period is approximately hundred dollars Do not round until the final answer Then round to two decimal places as needed

The right sum R is used to approximate the definite integral to the indicated accuracy How large must n be chosen to achieve the indicated accuracy The function is increasing over the interval 3 S In xdx R 0 05 n Round to the nearest integer as needed 8

Find the shaded area under the cosine curve in the figure to the right The prop under the curve in the figure is 1 0 0 y cos x HIN 2 X

The rate of healing A t in square centimeters per day for a certain type of abrasive skin wound is given approximately by the table CE t 0 A t 0 99 t 4 A t 0 62 1 0 83 2 0 73 3 0 68 5 6 7 0 55 0 47 0 42 A Use left and right sums over five equal subintervals to approximate the area under the graph of A t from t 0 t 5

Calculate the indicated Riemann sum S5 for the function f x 22 5x Partition 3 7 into five subintervals of equal length and for each subinterval XK 1Xxk let CK Xk 1 Xk 2 S5

Calculate 7x dx given the following 1 9 x dx 723 1 10 7x dx x dx 271 3 9 9 Sxd 1 10 73 1 Type an integer or a simplified fraction xdx 40

Use a graphing calculator to graph the region R bounded by the graphs of the indicated equations Use approximation echniques to find intersection points correct to two decimal places Describe R in set notation with double inequalities and evaluate the indicated integral correct to two decimal places y ex y 5 3x O A Graph the region R bounded by the graphs of the indicated equations Choose the correct answer below 20 R 14 4y dA OU OB 14 1 Q OU O C AY 14 4 3 Q Q D AY 20 14 Q S

Reverse the order of integration for the integral Then evaluate the integral with the order reversed 4 16 S S 0 4x 1 y dy dx 11 0 4x dx dy 2 1 y

Graph the region R bounded by the graphs of the given equations Use set notation and double inequalities to describe R as a regular x region and as a regular y region y 16 x y 0 0 x 4 Choose the correct graph of the region R below O A 20 B 20 Q O C Q 2 O D Q O 5

Use the description of the region R to evaluate the indicated integral Sex y dA R x y xsy x 5 x 8 R SS x dA Sexty

Find the volume of the solid under the graph of f x y over the region R bounded by the graphs of the indicated equations Sketch the region R The solid does not have to be sketched f x y 11 x y R is the region bounded by the graphs of x y 11 y 0 x 0 Sketch the region R Choose the correct graph below OA 12 X 12 Q OB 12 12 Q Q OC 12 12 O ON OD 12 X 12 Q

Reverse the order of integration for the following integral Evaluate the integral with the order reversed Do not attempt to evaluate the integral in the original form 66 S S 4yet 0 2 Reverse the order of integration Choose the correct reversed integral below dx dy 66 OA 4ye dy dx SS 00 OB OD 6 x SS 4ye dx dy 0 0 66 OC SS 4yex 00 dx dy 6 x SS 4ye dy dx 00

Find the average value of the function f x y x y over the rectangle R x y 0 x 1 0 y 1 The average value is Simplify your answer

Evaluate the double integral Select the order of integration carefully the problem is easy to do one way and difficult the other 4y SS R 5xy4 1 x 2 4y 5xy4 dA R x y 0 x 4 1sys1

Use both orders of iteration to evaluate the double integral SSxy da R ydA R x y 0 x 4 0 y 8 What is the iterated integral that is represented by integrating with respect to x first then y

Evaluate the iterated integral 15x y4dy dx 4 1 S S 15x y dy dx 3 0 3 0

Find the antiderivative Then use the antiderivative to evaluate the definite integral A S 2x 2y 4 dx 2 B 2x 2y 4 dx S 4 B A 2x 2y 4 dx x 2xy 4x E y Use C x to represent the arbitrary function of x or E y to represent the arbitrary function of y 2 2x 2y 4 dx

Under ideal conditions if a person driving a car slams on the brakes and skids to a stop the length of the skid marks in feet is given by the following formula where x is the weight of the car in pounds and y is the speed of the car in miles per hour L 0 0000133xy What is the average length of the skid marks for cars weighing between 2 200 and 2 900 pounds and traveling at speeds between 45 and 55 miles per hour Set up a double integral and evaluate it The average length of the skid marks is ft Do not round until the final answer Then round to two decimal places as needed

If an industry invests x thousand labor hours 11 x 21 and Sy million 1 y 2 in the production of N thousand units of a certain item then N is given by the following formula N x y x0 80y0 20 What is the average number of units produced for the indicated ranges of x and y Set up a double integral and evaluate it The average number of units produced for the indicated ranges of x and y is approximately Do not round until the final answer Then round to three decimal places as needed thousand units

The Cobb Douglas production function for a particular product is N x y 40x0 6 0 4 where x is the number of units of labor and y is the number of units of capital required to produce N x y units of the produ Each unit of labor costs 80 and each unit of capital costs 120 Answer the questions A and B below GECCEN A If 600 000 is budgeted for production of the product determine how that amount should be allocated to maximize production and find the maximum production Production will be maximized when using units of labor and units of capital

Find the values of x y and z that maximize xyz subject to the constraint 192 x 4y 16z 0

Evaluate the following and use Mathematica to verify each 11 a lim x limfx y dxdy 00 b 88 SS 1 1 x y 00 y dxdy

me of the solid A x 2 y x 0 y 9 about the y axis by rotating the region bounded by 429 disk washer or shell x 2 5

11 Find the volume of the solid obtained by rotating the region bounded by the following disk washer or shell A x 2 y x 0 y 9 about the y axis 429 x 2 5

and 0 1 and sin B and find the following trig ratios Remember a picture of the angles is part of your work 2 c sin 0 and cos B 7 Given cos 0 11

what does the following integral equal 6 d t dt 5 6 d t dt 12 and 5 L a 1 d t dt 8 2

where c 10 8 6 m 2 4 6 8 10 12 The net signed area in the above graph can be written as d and b 1 2 3 d mx b e C

Use the following figure which shows a graph of f x to find each of the indicated integrals a Note that the first area shaded red is 21 and the second shaded blue is 7 a f x dz b f x dx c f1 z dz S f x dx d

Use the graph of f x shown below to find the following integrals 2 2 a f x b If the red shaded area in the graph has area A estimate f x dx f x dx Your answer may be written in terms of A

3 5 Suppose f x dx 9 35 x dx 4 f x dx a 3 f x dx 3 5 b 6f x 8 dx

what does the following integral equal 41 6f x 3g x h x dx 59 41 59 41 59 41 f x dx 10 and g x dx 13 and h x dx 20

hat 1 2 dz 51 53 f x dx 1 11 Given that answer what is f f u du

4 Partition the given interval into the indicated number of subintervals 1 4 6

2 The position of an object at time t is given by s t Find the instantaneous velocity at the indicated value of t a s 1 3r 5 at 7 4 at 2

3 Let h 0 be a fixed real number Consider the elliptic paraboloid given by the equation 4 8 9 0 h a Show that the horizontal cross section of the solid at height 2 for 0 h is an ellipse and calculate its area A z b Calculate the volume of the elliptic paraboloid

Find the volume of the solid that lies within the sphere y 16 above the zy plane and outside the cone 6