Section Volumes by Slicing

Slides:

Advertisements

Similar presentations

Section Volumes by Slicing

Advertisements

Volumes by Slicing: Disks and Washers

DO NOW: Find the volume of the solid generated when the

Disks, Washers, and Cross Sections Review

Section Volumes by Slicing

More on Volumes & Average Function Value. Average On the last test (2), the average of the test was: FYI - there were 35 who scored a 9 or 10, which means.

Applications of Integration Copyright © Cengage Learning. All rights reserved.

Volumes – The Disk Method Lesson 7.2. Revolving a Function Consider a function f(x) on the interval [a, b] Now consider revolving that segment of curve.

6.2 - Volumes. Definition: Right Cylinder Let B 1 and B 2 be two congruent bases. A cylinder is the points on the line segments perpendicular to the bases.

7.1 Area Between 2 Curves Objective: To calculate the area between 2 curves. Type 1: The top to bottom curve does not change. a b f(x) g(x) *Vertical.

Volume: The Disk Method

TOPIC APPLICATIONS VOLUME BY INTEGRATION. define what a solid of revolution is decide which method will best determine the volume of the solid apply the.

Volume. Find the volume of the solid formed by revolving the region bounded by the graphs y = x 3 + x + 1, y = 1, and x = 1 about the line x = 2.

S OLIDS OF R EVOLUTION 4-G. Disk method Find Volume – Disk Method Revolve about a horizontal axis Slice perpendicular to axis – slices vertical Integrate.

7.2: Volumes by Slicing – Day 2 - Washers Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 Little Rock Central High School,

7.1 – 7.3 Review Area and Volume. The function v(t) is the velocity in m/sec of a particle moving along the x-axis. Determine when the particle is moving.

3 3 3 Find the volume of the pyramid: Consider a horizontal slice through the pyramid. s dh The volume of the slice is s 2 dh. If we put zero at the top.

Integral calculus XII STANDARD MATHEMATICS. Evaluate: Adding (1) and (2) 2I = 3 I = 3/2.

Finding Volumes.

Volume of a Solid by Cross Section Section 5-9. Let be the region bounded by the graphs of x = y 2 and x=9. Find the volume of the solid that has as its.

5/19/2015 Perkins AP Calculus AB Day 7 Section 7.2.

Review: Volumes of Revolution. x y A 45 o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge.

V OLUMES OF SOLIDS WITH KNOWN CROSS SECTIONS 4-H.

Section 7.2 Solids of Revolution. 1 st Day Solids with Known Cross Sections.

A = x2 s2 + s2 = x2 2s2 = x2 s2 = x2/2 A = x2/2 A = ½ πx2

Do Now: p.381: #8 Integrate the two parts separately: Shaded Area =

Warm Up Show all definite integrals!!!!! 1)Calculator Active: Let R be the region bounded by the graph of y = ln x and the line y = x – 2. Find the area.

Do Now: #10 on p.391 Cross section width: Cross section area: Volume:

Chapter 7 Quiz Calculators allowed. 1. Find the area between the functions y=x 2 and y=x 3 a) 1/3 b) 1/12 c) 7/12 d) 1/4 2. Find the area between the.

Finding Volumes Disk/Washer/Shell Chapter 6.2 & 6.3 February 27, 2007.

Volume of Cross-Sectional Solids

7.3.3 Volume by Cross-sectional Areas A.K.A. - Slicing.

Volumes of Solids Solids of Revolution Approximating Volumes

Solids of Revolution Disk Method

Volume: The Disc Method

Let R be the region bounded by the curve y = e x/2, the y-axis and the line y = e. 1)Sketch the region R. Include points of intersection. 2) Find the.

Volumes By Cylindrical Shells Objective: To develop another method to find volume without known cross-sections.

Finding Volumes Chapter 6.2 February 22, In General: Vertical Cut:Horizontal Cut:

Volumes Lesson 6.2.

Volume: The Shell Method

Volumes by Slicing 7.3 Solids of Revolution.

Volumes by Slicing. disk Find the Volume of revolution using the disk method washer Find the volume of revolution using the washer method shell Find the.

Section Volumes by Slicing 7.3 Solids of Revolution.

Solids of Known Cross Section. Variation on Disc Method  With the disc method, you can find the volume of a solid having a circular cross section  The.

Ch. 8 – Applications of Definite Integrals 8.3 – Volumes.

Volume Find the area of a random cross section, then integrate it.

Volume of Regions with cross- sections an off shoot of Disk MethodV =  b a (π r 2 ) dr Area of each cross section (circle) * If you know the cross.

 The volume of a known integrable cross- section area A(x) from x = a to x = b is  Common areas:  square: A = s 2 semi-circle: A = ½  r 2 equilateral.

C.2.5b – Volumes of Revolution – Method of Cylinders Calculus – Santowski 6/12/20161Calculus - Santowski.

Calculus 6-R Unit 6 Applications of Integration Review Problems.

7.2 Volume: The Disk Method (Day 3) (Volume of Solids with known Cross- Sections) Objectives: -Students will find the volume of a solid of revolution using.

Volumes of Solids of Rotation: The Disc Method

The region enclosed by the x-axis and the parabola is revolved about the line x = –1 to generate the shape of a cake. What is the volume of the cake? DO.

7.2 Volume: The Disk Method

Warm-Up! Find the average value of

Volumes – The Disk Method

Cross Sections Section 7.2.

Volume by Cross Sections

6.4 Integration of exponential reciprocal of x and some trig functions

Volume by Cross-sectional Areas A.K.A. - Slicing

Applications Of The Definite Integral

Warm Up Find the volume of the following shapes (cubic inches)

Area & Volume Chapter 6.1 & 6.2 February 20, 2007.

Chapter 6 Cross Sectional Volume

Section Volumes by Slicing

Warm Up Find the volume of the following 3 dimensional shapes.

AP problem back ch 7: skip # 7

Presentation transcript:

Section 8.2 - Volumes by Slicing 7.3 Solids of Revolution

Find the volume of the solid generated by revolving the regions bounded by about the x-axis.

Find the volume of the solid generated by revolving the regions bounded by about the x-axis.

Find the volume of the solid generated by revolving the regions bounded by about the y-axis.

Find the volume of the solid generated by revolving the regions bounded by about the line y = -1.

CALCULATOR REQUIRED The volume of the solid generated by revolving the first quadrant region bounded by the curve and the lines x = ln 3 and y = 1 about the x-axis is closest to a) 2.79 b) 2.82 c) 2.85 d) 2.88 e) 2.91

CALCULATOR REQUIRED

CALCULATOR REQUIRED

CALCULATOR REQUIRED Cross Sections

Let R be the region marked in the first quadrant enclosed by the y-axis and the graphs of as shown in the figure below Setup but do not evaluate the integral representing the volume of the solid generated when R is revolved around the x-axis. R Setup, but do not evaluate the integral representing the volume of the solid whose base is R and whose cross sections perpendicular to the x-axis are squares.

CALCULATOR REQUIRED

NO CALCULATOR

NO CALCULATOR

CALCULATOR REQUIRED Let R be the region in the first quadrant under the graph of a) Find the area of R. The line x = k divides the region R into two regions. If the part of region R to the left of the line is 5/12 of the area of the whole region R, what is the value of k? Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the x-axis are squares.

Let R be the region in the first quadrant under the graph of a) Find the area of R.

Let R be the region in the first quadrant under the graph of The line x = k divides the region R into two regions. If the part of region R to the left of the line is 5/12 of the area of the whole region R, what is the value of k? A

Let R be the region in the first quadrant under the graph of Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the x-axis are squares.

Let R be the region in the first quadrant bounded above by the graph of f(x) = 3 cos x and below by the graph of Setup, but do not evaluate, an integral expression in terms of a single variable for the volume of the solid generated when R is revolved about the x-axis. Let the base of a solid be the region R. If all cross sections perpendicular to the x-axis are equilateral triangles, setup, but do not evaluate, an integral expression of a single variable for the volume of the solid.