6.2C Volumes by Slicing with Known Cross-Sections.

Slides:



Advertisements
Similar presentations
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.
Advertisements

Applications of Integration
APPLICATIONS OF INTEGRATION
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
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.
7.3 Day One: Volumes by Slicing Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 Little Rock Central High School, Little.
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.
Objective: SWBAT use integration to calculate volumes of solids
7.3 Volumes Quick Review What you’ll learn about Volumes As an Integral Square Cross Sections Circular Cross Sections Cylindrical Shells Other Cross.
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
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.
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 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.
Volume of Cross-Sectional Solids
7.3.3 Volume by Cross-sectional Areas A.K.A. - Slicing.
Volume Section 7.3a. Recall a problem we did way back in Section 5.1… Estimate the volume of a solid sphere of radius 4. Each slice can be approximated.
Warm Up. Volume of Solids - 8.3A Big Idea Just like we estimate area by drawing rectangles, we can estimate volume by cutting the shape into slices,
Solids of Revolution Disk Method
Volume: The Disc Method
Ch 7.3 Volumes Calculus Graphical, Numerical, Algebraic by
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.
Finding Volumes Chapter 6.2 February 22, In General: Vertical Cut:Horizontal Cut:
Volumes Lesson 6.2.
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.
6.2 Volumes on a Base.
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.
Volumes by Disks and Washers Or, how much toilet paper fits on one of those huge rolls, anyway??
Ch. 8 – Applications of Definite Integrals 8.3 – Volumes.
Volumes of Solids with Known Cross Sections
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.
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.
Areas and Volumes Gateway Arch, St. Louis, Missouri Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College Photo by.
SECTION 7-3-C Volumes of Known Cross - Sections. Recall: Perpendicular to x – axis Perpendicular to y – axis.
7.3a: Volumes Learning Goals ©2007 Roy L. Gover ( Use integration to calculate volumes of solids using the Disk and Washer Methods. Use.
 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.
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 is.
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.
Drill: Find the area in the 4 th quadrant bounded by y=e x -5.6; Calculator is Allowed! 1) Sketch 2) Highlight 3) X Values 4) Integrate X=? X=0 X=1.723.
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,
The Disk Method (7.2) February 14th, 2017.
8-3 Volumes.
Volumes of solids with known cross sections
Finding Volumes.
Cross Sections Section 7.2.
Volume by Cross Sections
3 Find the volume of the pyramid:
Volume: Disk and Washer Methods
7.3 Day One: Volumes by Slicing
Warmup 1) 2) 3).
Volume of Solids with Known Cross Sections
Volume by Cross-sectional Areas A.K.A. - Slicing
Applications Of The Definite Integral
Area & Volume Chapter 6.1 & 6.2 February 20, 2007.
8.3 Day One: Volumes by Slicing
7.3 Day One: Volumes by Slicing
Volume: Disk and Washer Methods
Presentation transcript:

6.2C Volumes by Slicing with Known Cross-Sections

Known Cross Sections Method Volume can be calculated by finding area of known geometric shapes and multiplying by thickness (dx). Here is an example of squares stacked on top of a circular region.

Visualizations Rectangular Cross-Sections Semicircular Cross-Sections Equilateral Triangle Cross-Sections

We can find the area of each cross section, then add an infinite number of infinitely thin cross sections. When we multiply by thickness, we have volume.

Examples Cross sections may be rectangles, semi-circles or triangles. The base of the solid may be a rectangle, circle, triangle or an irregular shape. Mathematica

Method of Slicing: 1 Sketch the base of the solid (including a typical slice) and a typical cross section. Find a formula for A(x) and multiply by dx for width. 2 3 Find the limits of integration. 4 Integrate V(x) to find volume.

Find the volume of the solid whose base is bounded by the circle x2+y2=4 with square cross sections perpendicular to the x-axis. y x 422/51(c)

Find the volume of the solid whose base is bounded by the circle x2+y2=4 with semicircular cross sections perpendicular to the x-axis y x 422/51(c)

Find the volume of the solid whose base is bounded by the circle x2+y2=4 with equilateral triangle cross sections perpendicular to the x-axis. x y

Find the volume of the solid formed with the region defined by and as the base and cross sections that are squares perpendicular to the base and the x-axis. 81/10 sq units

3 Find the volume of the pyramid: Consider a horizontal slice through the pyramid. The volume of the slice is s2dh. If we put zero at the top of the pyramid and make down the positive direction, then s=h. h s This correlates with the formula: dh 3

p Cavalieri’s Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume. Identical Cross Sections p

Ex. Find the volume of the solid whose base is a circle of radius 1 in the first quadrant and with square cross-sections  x-axis.

Ex. Find the volume of the solid whose base is bounded by the x-axis, the y-axis, x = 9, and with semi-circular cross-sections  x-axis.

Find the volume of the solid whose base is a circle of radius 1 centered at the origin and with isosceles right triangles cross-sections  x-axis.