Section 14.5 The Area Problem; The Integral.

Slides:



Advertisements
Similar presentations
Solid of Revolution Revolution about x-axis. What is a Solid of Revolution - 1 Consider the area under the graph of y = 0.5x from x = 0 to x = 1:
Advertisements

Volumes by Slicing: Disks and Washers
- Volumes of a Solid The volumes of solid that can be cut into thin slices, where the volumes can be interpreted as a definite integral.
 A k = area of k th rectangle,  f(c k ) – g(c k ) = height,  x k = width. 6.1 Area between two curves.
Solids of Revolution Washer Method
7.1 Areas Between Curves To find the area: divide the area into n strips of equal width approximate the ith strip by a rectangle with base Δx and height.
The Disk Method (7.2) April 17th, I. The Disk Method Def. If a region in the coordinate plane is revolved about a line, called the axis of revolution,
Section 6.1 Volumes By Slicing and Rotation About an Axis
Volume: The Disk Method
Section 6.2.  Solids of Revolution – if a region in the plane is revolved about a line “line-axis of revolution”  Simplest Solid – right circular cylinder.
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.
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.
6.2C Volumes by Slicing with Known Cross-Sections.
Section 7.2 Solids of Revolution. 1 st Day Solids with Known Cross Sections.
A solid of revolution is a solid obtained by rotating a region in the plane about an axis. The sphere and right circular cone are familiar examples of.
Numerical Integration In general, a numerical integration is the approximation of a definite integration by a “weighted” sum of function values at discretized.
Section 7.3 – Volume: Shell Method. White Board Challenge Calculate the volume of the solid obtained by rotating the region bounded by y = x 2, x=0, and.
Application of integration. G.K. BHARAD INSTITUTE OF ENGINEERING Prepared by :- (1) Shingala nital (2) Paghdal Radhika (3) Bopaliya Mamata.
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 BY DISK or disc BY: Nicole Cavalier & Alex Nuss.
Volume: The Disc Method
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, Disk and Washer Methods Limerick Nuclear Generating Station, Pottstown,
Feed Back After Test. Aims: To know what a volume of revolution is and learn where the formula comes from. To be able to calculate a volume 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.
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.
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.
Volumes 7.3. Finding Volume Using the Cross Section Think of a cross section as a thin slice of the object. For Example:
7.2 Volume: The Disc Method The area under a curve is the summation of an infinite number of rectangles. If we take this rectangle and revolve it about.
Disks, Washers and Shells Limerick Nuclear Generating Station, Pottstown, Pennsylvania Disk Method.
5.2 Volumes of Revolution: Disk and Washer Methods 1 We learned how to find the area under a curve. Now, given a curve, we form a 3-dimensional object:
The Unit Circle and Circular Functions Trigonometry Section 3.3.
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,
SECTION 4-3-B Area under the Curve. Def: The area under a curve bounded by f(x) and the x-axis and the lines x = a and x = b is given by Where and n is.
Solids of Revolution Revolution about x-axis. What is a Solid of Revolution? Consider the area under the graph of from x = 0 to x = 2.
MTH1170 Integrals as Area.
The Disk Method (7.2) February 14th, 2017.
Finding Volumes by Integration
Volumes of solids with known cross sections
Solids of Revolution Shell Method
Solids of Revolution Shell Method
Cross Sections Section 7.2.
Volume of Prisms OCR Stage 8.
Revolution about x-axis
7.2 Volume: The Disc Method The area under a curve
Area of a Region Between Two Curves (7.1)
APPLICATIONS OF INTEGRATION
( ) Part (a) Shaded area = x dx - e dx
Volumes of Solids of Revolution
Write out the form of the partial fraction decomposition of the expression. Do not determine the numerical values of the coefficients. {image} 1. {image}
Find volumes of prism; solve real-life problems. CC.SS.7.G.6
3. Volumes.
Chapter 7.2: Volume The Disk Method The Washer Method Cross-sections
6.2 Volumes If a region in the plane is revolved about a line, the resulting solid is called a solid of revolution, the line is called the axis of revolution.
Calculus II (MAT 146) Dr. Day Monday, January 29, 2018
Volume of Solids with Known Cross Sections
Applications Of The Definite Integral
Volume - The Disk Method
Section 7.2 Day 1 Disk method
Integration Volumes of revolution.
INTEGRATION APPLICATIONS 2
6.1 Areas Between Curves To find the area:
Section 7.2 Day 2 Disk Method
Area of a Surface of Revolution
6.3 – Volumes By Cylindrical Shells
Volume of Prisms OCR Stage 8.
Presentation transcript:

Section 14.5 The Area Problem; The Integral

How is it calculated - 2 Just like the area under a continuous curve can be approximated by a series of narrow rectangles, the volume of a solid of revolution can be approximated by a series of thin circular discs: we could improve our accuracy by using a larger and larger number of circular discs, making them thinner and thinner

Volume of Revolution Formula The volume of revolution about the x-axis between x=a and x=b is: This formula you do need to know Think of is as the um of lots of circles … where area of circle = r2