Post on 08-Oct-2019
1
The area between two curves The Volume of the Solid of revolution (by slicing)
1. AREA BETWEEN the CURVES
ππππ = οΏ½οΏ½ ππππππππππ ππππππππππππππππ
οΏ½ β οΏ½ ππππππππππ ππππππππππππππππ
οΏ½οΏ½ dx
ππ = οΏ½ππππ = οΏ½[π¦π¦1(π₯π₯) β π¦π¦2(π₯π₯)]πππ₯π₯ππ
ππ
ππ
ππ
ππ = οΏ½ππππ = οΏ½[π₯π₯1(π¦π¦) β π₯π₯2(π¦π¦)]πππ¦π¦ππ
ππ
ππ
ππ
EX: Determine the area of the region bounded by y = 2x
2 +10 and y = 4x +16 between x = β 2 and x = 5
ππ = β« ππππ = β« οΏ½οΏ½ππππππππππ
πππππππππππππππποΏ½ β οΏ½ πππππππππππππππππππππππππποΏ½οΏ½ πππ₯π₯
ππππ
ππππ
2 EX: Determine the area of the region enclosed by y = sin x and y = cos x and the y -axis for 0 β€ π₯π₯ β€ ππ
2
EX: Determine the area of the enclosed area by π₯π₯ = 12π¦π¦2 β 3 and π¦π¦ = π₯π₯ β 1
Intersection: (-1,-2) and (5,4).
THE SAME: Determine the area of the enclosed area by π₯π₯ = 12π¦π¦2 β 3 and π¦π¦ = π₯π₯ β 1
So, in this last example weβve seen a case where we could use either method to find the area. However, the second was definitely easier.
3 EX: Find area Intersection points are: y = - 1 y = 3
ππ = οΏ½[(βπ¦π¦2 + 10) β (π¦π¦ β 2)2]πππ¦π¦3
β1
= οΏ½[β2π¦π¦2 + 4π¦π¦ + 6]πππ¦π¦3
β1
= οΏ½β23π¦π¦3 + 2π¦π¦2 + 6π¦π¦οΏ½
3
β1
ππ =643
Volume of REVOLUTION
βͺ Find the Volume of revolution using the disk method βͺ Find the volume of revolution using the washer method βͺ Find the volume of revolution using the shell method βͺ Find the volume of a solid with known cross sections
Area is only one of the applications of integration. We can add up representative volumes in the same way we add up representative rectangles. When we are measuring volumes of revolution, we can slice representative disks or washers.
DISK METHOD
ππ = οΏ½ππππ
ππ
ππ
ππππ = ππππ2πππ₯π₯
ππππ = ππ[ππ(π₯π₯)]2πππ₯π₯
ππππ = ππ[ππ(π₯π₯) + ππ]2πππ₯π₯
ππππ = ππ[ππ(π₯π₯) β ππ]2πππ₯π₯
ππππ = ππ[ππ β ππ(π₯π₯)]2πππ₯π₯
4
ππ = οΏ½ππππππ
ππ
WASHER METHOD
A solid obtained by revolving a region around a line.
5
6 Volumes by Cylindrical Shells
7
8
o Procedure: volume by slicing sketch the solid and a typical o cross section find a formula for the area, A(x), of the o cross section find limits of integration integrate o A(x) to get volume Find the volume of a solid whose base is the circle x2 + y2 = 4 and where cross sections perpendicular to the x-axis are a) squares b) Equilateral triangles c) semicircles
ππππ = ππ πππ₯π₯ ππ = ππ2
ππ = 4 οΏ½(4 β π₯π₯2)πππ₯π₯ =128
3
2
β2
x2 + y2 = 4 π¦π¦ = β4 β π₯π₯2
length of a side is : 2οΏ½4β π₯π₯2
ππ = 12ππ οΏ½ππ2 β οΏ½
ππ2οΏ½2
=β34
ππ2 = β3(4 β π₯π₯2)
ππ = οΏ½β3(4 β π₯π₯2)πππ₯π₯ =32β3
β 18.4752
β2
x2 + y2 = 4 π¦π¦ = β4 β π₯π₯2
ππ = 12
ππ οΏ½ππ2οΏ½2
=18ππ ππ2 = ππ
4 β π₯π₯2
2
ππππ = ππ πππ₯π₯
ππ = οΏ½ππ 4 β π₯π₯2
2πππ₯π₯
2
β2
=16ππ
3β 16.755
x2 + y2 = 4 π¦π¦ = β4 β π₯π₯2
9 d) Isosceles right triangles
The base of the volume of a solid is the region bounded by the curve ππ(π₯π₯) = 4π₯π₯ β π₯π₯2 ππππππ ππ(π₯π₯) = π₯π₯2 . Find the volumes of the solids whose cross sections perpendicular to the x-axis are the following:
ππ = 12ππ
ππ2
tanππ 4οΏ½=
ππ2
4= 4 β π₯π₯2
ππππ = ππ πππ₯π₯
ππ = οΏ½(4 β π₯π₯2) πππ₯π₯ =323β 10.667
2
β2
x2 + y2 = 4 π¦π¦ = β4 β π₯π₯2
10 PRACTICE: 1. Find the volume of the solid generated by revolving about the x-axis the region bounded by the graph of ππ(π₯π₯) = βπ₯π₯ β 1 the x-axis, and the line x = 5. Draw a sketch. 1. ANS: 8Ο
2. Find the volume of the solid generated by revolving about the x -axis the region bounded by the graph
of π¦π¦ = βcosπ₯π₯ where 0 β€ π₯π₯ β€ ππ2
the x-axis, and the y-axis. Draw a sketch. 2. ANS: Ο
3. Find the volume of the solid generated by revolving about the y-axis the region in the first quadrant bounded by the graph of y = x2, the y-axis, and the line y = 6. Draw a sketch. 3. ANS: 18 Ο
4. Using a calculator, find the volume of the solid generated by revolving about the line y = 8 the region bounded by the graph of y = x 2 + 4, the line y = 8. Draw a sketch. ANS: 512/15 Ο
5. Using a calculator, find the volume of the solid generated by revolving about the line y = β3 the region bounded by the graph of y = ex, the y-axis, the lines x = ln 2 and y = β 3. Sketch. 5. ANS: 13.7383 Ο
6. Using the Washer, find the volume of the solid generated by revolving the region bounded by y = x 3 and y = x in the first quadrant about the x-axis. Draw a sketch. Method (just a fancy name β use sketch and common sense!!! instead of given boundaries, you have to find it as intersection of two curves and then use sketch to subtract one volume from another ) 6. ANS: 4Ο/21
7. Using the Washer Method and a calculator, find the volume of the solid generated by revolving the region bounded by y = x 3 and y = x about the line y = 2. Draw a sketch. 7. ANS: 17Ο/21
8. Using the Washer Method and a calculator, find the volume of the solid generated by revolving the region bounded by y = x2 and x = y2 about the y-axis. Draw a sketch. 8. ANS: 3Ο/10
AGAIN PRACTICE:
1. The base of a solid is the region enclosed by the ellipse π₯π₯2
4+ π¦π¦
2
25 = 1
. The cross sections are perpendicular to the x-axis and are isosceles right triangles whose hypotenuses are on the ellipse. Find the volume of the solid. 1. ANS: V = 200/3
2. The base of a solid is the region enclosed by a triangle whose vertices are (0, 0), (4, 0) and (0, 2). The cross sections are semicircles perpendicular to the x-axis. Using a calculator, find the volume of the solid.
11 2. ANS: V = 2.094
3. Find the volume of the solid whose base is the region bounded by the lines x + 4 y = 4, x = 0, and y = 0, if the cross sections taken perpendicular to the x-axis are semicircles.
3. ANS: V = Ο/6
4. The base of a solid is the region in the first quadrant bounded by the y-axis, the graph of y = arctanx, the horizontal line y = 3, and the vertical line x = 1. For this solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid? 4. ANS: V = β«βΒΉ (3 - arctan(x))Β² dx = 6.61233
5. A solid has its base is the region bounded by the lines x + 2y = 6, x = 0 and y = 0 and the cross sections taken perpendicular to x-axis are circles. Find the volume the solid.
5. ANS: 9/2 Ο
6. A solid has its base is the region bounded by the lines x + y = 4, x = 0 and y = 0 and the cross section is perpendicular to the x-axis are equilateral triangles. Find its volume. 6. ANS: V = 16β3/3