Chapter 3 Integration PDF

Preview

Summary

Download Chapter 3 Integration PDF

Description

3.1 Double Integrals Focus of Attention  Definition as a limit of Riemann sum – interpretation as volume under graph  How are double integrals evaluated as iterated integrals?  Does the order of integration matter?  How are the limits of integration determined?  How are double integrals used to calculate volume of solid & area of plane region 3.2 Iterated Integrals 3.2.1 Double Integration Over Rectangular Region Illustration • Evaluating double integrals as iterated integrals • Setting up in rectangular coordinates – region of integration, order of integration, limits of integration 1

Question 1 Evaluate the integrals. 3 2

(a)

∫ ∫ (1 + 8xy ) dydx 0 1 2 3

(b)

∫ ∫ (1 + 8xy ) dxdy 1 0

Prompts/Questions • Which theorem do you use to evaluate the integrals? • What is the inner integral? ○ Which variable is kept fixed? • What is the outer integral? • What basic integration formulas do you know? 2

Question 2 Compute ∫∫ (2 − y )dA where R is a rectangle R

with vertices (0, 0), (3, 0), (3, 2) and (0, 2).

Prompts/Questions • What do you do to write the integral as an iterated integral? o What is the region of integration? o What are the limits of integration?

• Does the order of integration matter? 3

Question 3 Evaluate

∫∫ x cos xy d A for R

R:0 ≤x ≤

π 2

, 0 ≤ y ≤ 1.

Prompts/Questions • Sketch the region R over which the integration is to be performed? • How do you write the integral as an iterated integral? o Which order of integration do you choose? o How do you find the limits of integration? • Do you need to use any basic integration formulas? 4

Reflection • How are double integrals evaluated as iterated integrals? …………………………………………………… ……………………………………………… • How do you decide on the order of integration? ………………………………………………… • What do you do to determine the limits of integration? …………………………………………………… • What do you notice about the limits of double integrals over rectangular region? …………………………………………………

3.2.2 Double Integration Over Nonrectangular Regions Illustration • Evaluating double integrals as iterated integrals • Setting up in rectangular coordinates – region of integration, order of integration, limits of integration • Reversing the order of integration 5

Question 1 Evaluate ∫∫ (x + y ) d A over the region R R

enclosed by the lines y = 0 , y = 2x and x = 1.

Prompts/Questions • Write the integral as an iterated integral. ○ On which plane is the region of integration? Sketch it. ○ Which order of integration is more convenient? • What are the limits of integration? 6

Question 2 Evaluate the integral by reversing the order of integration. 1 1

2 xy y ∫ ∫ e dydx 0 x

Prompts/Questions • Which part of the integral informs you about the region of integration? o Find and sketch the region. o Can you determine the limits of integration for the reversed order? • Why is it worthwhile to reverse the order of integration? 7

Question 3 4 2

y2

Evaluate ∫ ∫ e dydx . 0 x 2

Prompts/Questions • The integral is impossible to evaluate in the given order. Why? • What do you do to reverse the order of integration? o Can you find and sketch the region over which the integration is being performed? • How do you determine the limits of integration for the reversed order? 8

Question 4 Construct TWO examples of double integrals that are readily evaluated by integrating in one order but not in the reverse order.

Reflection

• How do you evaluate double integrals? ………………………………………………… ………………………………………………… • How do you choose the preferable order of integration? ………………………………………………… • How are the limits of integration determined? ………………………………………………… ………………………………………………… • What do you do to evaluate double integrals by reversing the order of integration? ………………………………………………… ………………………………………………… ………………………………………………… 9

3.2.3 Double Integral as Area and Volume Illustration Using double integral to find • area of a plane region • volume of a solid Question 1 Find the area of the region bounded by y = x and y = x 2 in the first quadrant.

Prompts/Questions • What is the formula for finding an area using a double integral? • How do you set up the iterated integral? o Sketch the region of integration. o Choose the order of integration. o Determine the limits. 10

Question 2 Find the area of the region enclosed by the parabola y = x 2 and the line y = x + 2 .

Prompts/Questions • How do you use double integral to find area? o State the formula. • What do you need to know to set up the integral? o What is the region of integration? Sketch it. o Which order is preferable? o What are the limits? 11

Question 3 Find the area of the region bounded by the graphs y 2 = 4 − x and y 2 = 4 − 4x .

Prompts/Questions Compare Q1, Q2 and Q3. • What do you do to find area using double integral? o What formula do you use? o How do you set up the integral? • How do you calculate the iterated integrals? 12

Question 4 Use a double integral to find the volume of the tetrahedron bounded by the coordinate planes and the plane z = 4 − 4x − 2y .

Prompts/Questions • What is the formula for finding volume using a double integral? • How do you set up the integral? o Sketch the surface. o On which plane is the region over which the integration is to be performed? What is the order of integration? o Determine the limits. 13

Question 5 Find the volume of the solid lying in the first octant and bounded by the graphs of z = 4 − x 2 − y 2 and y = 2 − 2x 2 .

Prompts/Questions • How do you find volume using double integral? o State the formula. • What do you need to know to set up the integral? o Can you sketch/imagine the bounding surfaces? o On which plane is the region of integration? o How do you find the limits of integration? 14

Question 6 Find the volume of the solid bounded by the cylinder x 2 + y 2 = 4 and the plane y + z = 4 and z = 0 .

Prompts/Questions Compare Q4, Q5 and Q6. • What do you do to find volume using double integral? o What formula do you use? o How do you set up the iterated integral? • How do you calculate the iterated integrals? 15

Reflection • What is the formula for finding an area using a double integral? A volume using a double integral? ………………………………………………… ………………………………………………… • What do you do to set up an iterated integral that gives you area? ………………………………………………… ………………………………………………… • What do you do to set up an iterated integral that gives you volume? ………………………………………………… …………………………………………………

16

3.3 Double Integral in Polar Form Focus of Attention  How do you change a double integral in rectangular coordinates into a double integral in polar coordinates?  What do you look for when considering using polar coordinates? Illustration • Evaluating polar double integrals • Transforming double integrals in rectangular coordinates to polar coordinates • Finding areas and volumes using polar double integrals

17

Question 1 Find the limits of integration for integrating f (r, θ ) over the region R that lies inside the cardiod r = 1 + cos θ and outside the circle r = 1.

Prompts/Questions • Can you identify and sketch the region of integration? o How do you find the r-limits of integration? The θ-limits? • How do you set up the polar iterated integral? • What is the order of integration? 18

Question 2 Evaluate ∫∫ (x 2 + y 2 + 1)d A where R is the R

region inside the circle x 2 + y 2 = 4 .

Prompts/Questions • Why are polar coordinates a better choice? • How do you transform the integral into polar iterated integral? o What is the order of integration? o What are the limits of integration? • What basic integration formulas do you know? 19

Question 3 Evaluate ∫∫ x dA where R is the region R

bounded above by the line y = x and below by the circle x 2 + y2 − 2y = 0.

Prompts/Questions • The integral would be easier to evaluate in polar coordinates. Why? • What do you do to transform the integral into polar iterated integral? o Can you sketch the region of integration? o How do you determine the limits of integration? • How is the integral calculated? 20

Question 4 Use polar double integral to find the area enclosed by the three-petal rose r = sin 3θ .

Prompts/Questions • Sketch the region. o Do you know the graph of the polar curve? o What do you notice about the shape? • What is the formula for area in polar coordinates? • How do you set up the polar iterated integral? o What is the preferable order of integration? • What are the limits of integration? 21

Question 5 Find the area bounded by the polar axis, part of the spiral rθ = 2 and between the graphs r = 2 and r = 3 .

Prompts/Questions

• What do you do to set up the polar iterated integral? o State the formula for area in polar coordinates. o Can you sketch the region of integration? o How do you choose the order of integration? • What are the limits of integration? 22

Question 6 Find the volume of the solid bounded by the given surfaces. (a) The cylinder x 2 + y 2 = 4 and the planes y + z = 4 and z = 0 . (b) The paraboloids z = 4 − x 2 − y 2 and z = x 2 + y2 .

Prompts/Questions • Can you sketch the bounding surfaces? o Identify and sketch the region of integration. • Why are polar coordinates a better choice? • How do you set up the iterated integral? o What formula do you use? o How do you change the variables from rectangular to polar coordinates? • How do you calculate the iterated integrals? 23

Reflection • How do you know a double integral would be easier to evaluate in polar coordinates? ………………………………………………… ………………………………………………… • How do you transform a double integral in rectangular coordinates into polar iterated integral? ………………………………………………… ………………………………………………… • How are the polar limits of integration determined? ………………………………………………… …………………………………………………

24

3.4 Triple Integrals Focus of Attention  Definition as a limit of Riemann sum – interpretation as volume of solid  How are triple integrals evaluated as iterated integrals?  Does the order of integration matter?  How are the limits of integration determined?  How is triple integral used to find volume?  How do you change a triple integral in rectangular coordinates into a triple integral in cylindrical coordinates or spherical coordinates?  What do you look for when considering using cylindrical coordinates or spherical coordinates? 3.4.1 Triple integrals in Rectangular/Cartesian Coordinates Illustration • Evaluating triple integrals as iterated integrals • Setting up integrals in rectangular coordinates – region of integration, order of integration, limits of integration 25

Question 1 Evaluate the integrals. 4 3 5

(a)

∫ ∫ ∫ dxdydz 1 −2 2

1 x 2 ln z

(b) ∫ ∫ ∫ xe y dydzdx 0 x

0

Prompts/Questions • Which theorem do you use to evaluate iterated integrals? • What is the inner integral? o Which variables are held fixed? • What is the middle integral? o Which variable is held fixed? • What basic integration formulas do you know? 26

Question 2 Evaluate ∫∫∫ z 2ye x dV , over the rectangular G

box G defined by 0 ≤ x ≤ 1, 1 ≤ y ≤ 2, − 1 ≤ z ≤ 1 .

Prompts/Questions • How do you write the integral as an iterated integral? o What is the region of integration? Sketch it. o What are the limits of integration? • Does the order of integration matter? • How are the iterated integrals calculated? 27

Question 3 Let G be the wedge in the first octant cut from the cylindrical solid y 2 + z 2 = 1 by the planes y = x and x = 0 . Evaluate ∫∫∫ z dV . G

Prompts/Questions • Sketch the solid region of integration. o Can you identify the type of region? • What do you do to write the integral as an iterated integral? o Which order of integration do you choose? o How do you determine the limits of integration for the innermost integral? The middle integral? The outer integral? • What basic integration formulas do you need? 28

Reflection • How do you set up a triple integral as an iterated integral? ………………………………………………… • How do you decide on the order of integration? ………………………………………………… • How do you determine the limits of integration? ………………………………………………… • How do you evaluate triple integrals as iterated integrals? ………………………………………………… • What do you notice about the limits of integration over a rectangular box? Nonrectangular region? …………………………………………………

3.4.2 Volume by Triple Integrals Illustration • Using a triple integral to find volume • Setting up integrals in rectangular coordinates – region of integration, order of integration, limits of integration 29

Question 1 The volume of a closed bounded region G in space is given by 1

0

y2

∫∫∫ dV = ∫ ∫ ∫ dzdydx G

0 −1 0

Rewrite the integral as an equivalent iterated integral in the order (a) dy dz dx (b) dx dy dz

Prompts/Questions • Which part of the integral informs you about the region? o Sketch the region whose volume is given by the integral. o Can you identify and sketch the region R on the xz-plane? The region R on the yz-plane? • How do you set up the equivalent integral? • What are the limits of the innermost integral? The limits of the middle integral? The outer integral? 30

Question 2 Find the volume of the region of the solid in the first octant bounded by the coordinate planes, the plane y + z = 2 and the cylinder x = 4 − y2 .

Prompts/Questions • What is the formula for finding volume using triple integral? o Sketch the solid. What are the bounding surfaces? • On which plane is the region R? Sketch it. • Which order of integration is convenient? • How do you set up the limits of the iterated integral? 31

Question 3 Find the volume of the region bounded above by z = 4 − x 2 − y 2 , below by z = 0 and laterally by x 2 + y 2 ≤ 1.

Prompts/Questions • How do you use triple integral to find volume? o State the formula. • How do you set up the iterated integral? o Can you sketch the solid region? o Which order of integration do you choose? o On which plane is the region R? o What are the limits of integration? • How do you calculate the iterated integral? • What integral rules do you use? 32

Reflection • What is the formula for finding a volume using triple integral? ………………………………………………… • What do you do to set up an iterated integral that gives you volume? ………………………………………………… ………………………………………………… • How is the iterated integral calculated? ………………………………………………… ………………………………………………… 3.4.3 Cylindrical Coordinates System Illustration Integration in cylindrical coordinates • Evaluating triple integrals using cylindrical coordinates • Setting up integrals in cylindrical coordinates – region of integration, order of integration, limits of integration • Finding volumes using cylindrical coordinates 33

Question 1 Use cylindrical coordinates to evaluate 3

∫

9−x 2 9−x 2 −y 2

∫

−3 − 9−x 2

∫

x 2 dz dy dx

0

Prompts/Questions • Which part of the integral informs you about the region of integration? Sketch it. • How do you transform the integral into cylindrical coordinates? o Integrand o Differential of integration (dV) o Domain/region of integration • How is the iterated integral calculated? • What basic integration formulas do you use? 34

Question 2 Find the limits of integration in cylindrical coordinates for integrating a function f (r , θ , z ) over the region G bounded below by the plane z = 0 , laterally by the circular cylinder x 2 + y2 = 2y and above by the paraboloid z = x 2 + y 2.

Prompts/Questions • Can you identify and sketch the region of integration? • How do you set up the iterated integral? o What is the relation between Cartesian and cylindrical coordinates? • How do you find the r-limits of integration? The θ-limits? The z-limits? 35

Question 3 Find the volume of the region bounded above by z = 4 − x 2 − y 2 , below by z = 0 and laterally by x 2 + y 2 ≤ 1.

Prompts/Questions • What is the region of integration? o Identify and sketch the bounding surfaces. • Why are cylindrical coordinates a better choice? o How do you describe the region in cylindrical coordinates? • How do you set up the iterated integral? o What is the formula for finding volume? o What are the limits of integration? • How do you calculate the iterated integral? 36

Question 4 Find the volume of solid in the first octant that is bounded by the cylinder x 2 + y 2 = 2y , by the cone z =

x 2 + y 2 and the xy-plane.

Prompts/Questions • How do you use triple integral to find volume? o State the formula. • Would you consider using cylindrical coordinates? • What do you do to set up the iterated integral? ○ Sketch the region of integration. • Find the limits of integration. 37

Reflection • What do you look for when considering using cylindrical coordinates? ………………………………………………… ………………………………………………… • How do you transform a triple integral in rectangular coordinates into cylindrical iterated integral? ………………………………………………… ………………………………………………… ………………………………………………… ………………………………………………… • How are the limits of integration determined in cylindrical coordinates? ………………………………………………… ………………………………………………… • How do you use a triple integral in cylindrical coordinates to calculate volume? ………………………………………………… …………………………………………………

38

3.4.4 Spherical Coordinates System Illustration Integration in spherical coordinates • Evaluating triple integrals using spherical coordinates • Setting up integrals in spherical coordinates – region of integration, order of integration, limits of integration • Finding volumes using spherical coordinates

39

Question 1 Use spherical coordinates to evaluate 2

∫

4−x 2

∫

−2 − 4−x 2

4−x 2 −y 2

∫

z 2 x 2 + y 2 + z 2 dz dy dx

0

Prompts/Questions • What do the limits of integration tells you about the solid? Sketch the solid. • How do you change the integral to spherical coordinates? o Integrand o Volume element o Limits of integration • Compare the integrals in rectangular and spherical coordinates. • When would you consider using spherical coordinates? 40

Question 2 Let G be the region bounded below by the cone z = x 2 + y 2 and above by the plane z = 1. Set up the triple integrals in spherical coordinates that give the volume of G using the following orders of integration. (a) d ρ dφ dθ (b) dφ d ρ dθ

Prompts/Questions • How do you set up the iterated integral? o Can you sketch the solid? o What is the relation between Cartesian and spherical coordinates? • How do you determine the ρ -limits of integration? The φ-limits? The θ-limits? 41

Question 3 Let G be the region bounded below by the plane z = 0, above by the sphere x2 + y2 + z2 = 4 and on the sides by the cylinder x 2 + y 2 = 1. SET UP the triple integrals in spherical coordinates that give the volume of G.

Prompts/Questions • What is the region of integration? o Identify and sketch the bounding surfaces. • What is the formula for finding volume using a triple integral? • Why are spherical coordinates a better choice? o How do you describe the region in spherical coordinates? • How do you set up the iterated integral? o Which order of integration is preferable? o What are the limits of integration? 42

Question 4 Find the volume of the “ice cream ...