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

Slides:

Advertisements

Similar presentations

Volumes by Slicing: Disks and Washers

Advertisements

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.

VOLUMES Volume = Area of the base X height. VOLUMES.

 A k = area of k th rectangle,  f(c k ) – g(c k ) = height,  x k = width. 6.1 Area between two curves.

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.

Section 6.1 Volumes By Slicing and Rotation About an Axis

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

Chapter 6 – Applications of Integration

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.

SECTION 7.3 Volume. VOLUMES OF AN OBJECT WITH A KNOWN CROSS-SECTION  Think of the formula for the volume of a prism: V = Bh.  The base is a cross-section.

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

7.3 Day One: Volumes by Slicing Find the volume of the pyramid: Consider a horizontal slice through the pyramid. s dh The volume of the slice.

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.

7.3 Volumes Quick Review What you’ll learn about Volumes As an Integral Square Cross Sections Circular Cross Sections Cylindrical Shells Other Cross.

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.

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.

Section Volumes by Slicing

6.2C Volumes by Slicing with Known Cross-Sections.

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

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, B Volumes by the Washer Method Limerick Nuclear Generating Station,

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

7.3 VOLUMES. Solids with Known Cross Sections If A(x) is the area of a cross section of a solid and A(x) is continuous on [a, b], then the volume of the.

Application of integration. G.K. BHARAD INSTITUTE OF ENGINEERING Prepared by :- (1) Shingala nital (2) Paghdal Radhika (3) Bopaliya Mamata.

7.3 Day One: Volumes by Slicing. Volumes by slicing can be found by adding up each slice of the solid as the thickness of the slices gets smaller and.

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

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.

Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

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

Volumes Lesson 6.2.

Section 7.3 Volume: The Shell Method. When finding volumes of solids by the disk (or washer) method we were routinely imagining our ‘slices’ under the.

Volumes by Slicing 7.3 Solids of Revolution.

Finding Volumes. In General: Vertical Cut:Horizontal Cut:

6.3 Volumes of Revolution Tues Dec 15 Do Now Find the volume of the solid whose base is the region enclosed by y = x^2 and y = 3, and whose cross sections.

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.

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

Chapter Area between Two Curves 7.2 Volumes by Slicing; Disks and Washers 7.3 Volumes by Cylindrical Shells 7.4 Length of a Plane Curve 7.5 Area.

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.

6.2 - Volumes Roshan. What is Volume? What do we mean by the volume of a solid? How do we know that the volume of a sphere of radius r is 4πr 3 /3 ? How.

SECTION 7-3-C Volumes of Known Cross - Sections. Recall: Perpendicular to x – axis Perpendicular to y – axis.

 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.

Overview Basic Guidelines Slicing –Circles –Squares –Diamonds –Triangles Revolving Cylindrical Shells.

6.3 Volumes by Cylindrical Shells. Find the volume of the solid obtained by rotating the region bounded,, and about the y -axis. We can use the washer.

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.

Sec 6.2: VOLUMES Volume = Area of the base X height.

Extra Review Chapter 7 (Area, Volume, Distance). Given that is the region bounded by Find the following  Area of  Volume by revolving around the x-axis.

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 SOLIDS OF REVOLUTION Rizzi – Calc BC. UM…WHAT?  A region rotated about an axis creates a solid of revolution  Visualization Visualization.

Section 7.3: Volume The Last One!!! Objective: Students will be able to… Find the volume of an object using one of the following methods: slicing, disk,

Finding Volumes Chapter 6.2 February 22, 2007.

Warm up Find the area of surface formed by revolving the graph of f(x) = 6x3 on the interval [0, 4] about the x-axis.

Applications Of The Definite Integral

Section Volumes by Slicing

6.3 – Volumes By Cylindrical Shells

Presentation transcript:

Ch. 8 – Applications of Definite Integrals 8.3 – Volumes

The volume of a solid with a known, integrable cross-sectional area of A(x) from x=a to x=b is Ex: Find the volume of the solid formed by revolving the region of the curve between the x-axis and y = 2 + cosx around the x-axis over [0, 2π]. –Graph the function from [0, 2π]. Picture the shape visually. –The revolution of this shape will produce an hourglass-like shape –The formula above says we need a cross-sectional area formula, and since the cross-section is a circle, here’s our formula: –Now integrate over the proper interval. Use your calculator. r

Ex: The region enclosed by the y-axis and the graphs of y = x + 2 and y = 4 – x 2 is revolved around the x-axis to form a solid. Find its volume. –Find a cross-sectional area formula: –What are the limits of integration? –Now integrate over the proper interval. Use your calculator. r1r1 r2r2 This is called the washer method because the cross- section is a washer

Ex: The region enclosed by the graphs of and y = x + 3 is revolved around the x-axis to form a solid. Find its volume. NO CALCULATOR! –Find a cross-sectional area formula: –What are the limits of integration? –Now integrate over the proper interval. r1r1 r2r2

Ex: The region enclosed by the x-axis and the graph of is revolved around the line y = 3 to form a solid. Find its volume. NO CALCULATOR! –Use the washer method, but the washer radii will be trickier to find: –Whew! Limits of integration: –Now integrate over the proper interval.

Ex: The region enclosed by the x-axis and the graph of over [1, 4] is revolved around the y-axis to form a solid. Find its volume. –When the radius of the cross-section is parallel to the axis of revolution, use the shell method! r(x) is the radius as a function of x h(x) is the height as a function of x –Find radius and height formulas: –Now integrate over the proper interval.

Ex: The region enclosed by the line x = 1, the x-axis, and the graph of is revolved around the line x = 3 to form a solid. Find its volume. NO CALCULATOR! –Use the shell method, but watch out for the tricky radius! –Find radius and height formulas: –Now integrate over the proper interval. x3 – x

Ex: A region is bounded by the x-axis, x = 3, and. This region is cut into square cross-sections that sit perpendicular to the x-axis. Find the volume of the solid generated by these cross- sections without a calculator. –I’ll cheat and let you look at my calculator for the graph. –We need a cross-sectional area formula for a square: –Now integrate over the proper interval.

Ex: A region is bounded by the x-axis, x = 3, and. This region is cut into equilateral triangle cross-sections that sit perpendicular to the x-axis. Find the volume of the solid generated by these cross-sections without a calculator. –I’ll cheat and let you look at my calculator for the graph. –Know this formula for equilateral triangle cross-sections: –Now integrate over the proper interval.