Calculus Notes Ch 6.2 Volumes by slicing can be found by adding up each slice of the solid as the thickness of the slices gets smaller and smaller, in.

Slides:

Advertisements

Similar presentations

Volumes by Slicing: Disks and Washers

Advertisements

Volumes using washers. Now that you have successfully designed a 4 by 4 meter nose cone, your boss brings to you a larger nose cone that is 16 meters.

Disk and Washer Methods

- 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.

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.

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, day 2 Disk and Washer Methods Limerick Nuclear Generating Station,

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

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.

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.

Applications of Integration

4/30/2015 Perkins AP Calculus AB Day 4 Section 7.2.

APPLICATIONS OF INTEGRATION Volumes by Cylindrical Shells APPLICATIONS OF INTEGRATION In this section, we will learn: How to apply the method of.

The Shell Method Volumes by Cylindrical Shells By Christine Li, Per. 4.

Section 6.1 Volumes By Slicing and Rotation About an Axis

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.

Review Problem – Riemann Sums Use a right Riemann Sum with 3 subintervals to approximate the definite integral:

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.

Chapter 6 – Applications of Integration

 Find the volume of y= X^2, y=4 revolved around the x-axis  Cross sections are circular washers  Thickness of the washer is xsub2-xsub1  Step 1) Find.

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

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

Volume: The Shell Method Lesson 7.3. Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal.

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

7.3 Volumes by Cylindrical Shells

MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.3 – Volumes by Cylindrical Shells Copyright © 2006 by Ron Wallace,

7.2 Volumes APPLICATIONS OF INTEGRATION In this section, we will learn about: Using integration to find out the volume of a solid.

(SEC. 7.3 DAY ONE) Volumes of Revolution DISK METHOD.

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

7.3 day 2 Disks, Washers and Shells Limerick Nuclear Generating Station, Pottstown, Pennsylvania.

6.3 Volumes by Cylindrical Shells APPLICATIONS OF INTEGRATION In this section, we will learn: How to apply the method of cylindrical shells to find out.

Chapter 6 – Applications of Integration 6.3 Volumes by Cylindrical Shells 1Erickson.

Volumes of Revolution Disks and Washers

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.

Solids of Revolution Disk Method

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, Volumes of rotation by Disks Limerick Nuclear Generating Station,

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,

Ch 7.3 Volumes Calculus Graphical, Numerical, Algebraic by

+ C.2.5 – Volumes of Revolution Calculus - Santowski 12/5/2015Calculus - Santowski.

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.

Disks, Washers and Shells Limerick Nuclear Generating Station, Pottstown, Pennsylvania.

Augustin Louis Cauchy 1789 – 1857 Augustin Louis Cauchy 1789 – 1857 Cauchy pioneered the study of analysis, both real and complex, and the theory of permutation.

Clicker Question 1 What is the area enclosed by f(x) = 3x – x 2 and g(x) = x ? – A. 2/3 – B. 2 – C. 9/2 – D. 4/3 – E. 3.

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.

Volume: The Disk Method. Some examples of solids of revolution:

Lecture 1 – Volumes Area – the entire 2-D region was sliced into strips Before width(  x) was introduced, only dealing with length ab f(x) Volume – same.

8.1 Arc Length and Surface Area Thurs Feb 4 Do Now Find the volume of the solid created by revolving the region bounded by the x-axis, y-axis, and y =

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.

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.

Volumes by Cylindrical Shells. What is the volume of and y=0 revolved around about the y-axis ? - since its revolving about the y-axis, the equation needs.

Calculus April 13Volume: the Shell Method. Instead of cutting a cylinder into disks (think of cutting an onion into slices), we will look at layers of.

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:

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.

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.

7.4 Day 2 Surface Area Greg Kelly, Hanford High School, Richland, Washington(Photo not taken by Vickie Kelly)

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.

Rotational Volumes Using Disks and Washers.

Volumes – The Disk Method

Volumes of Solids of Revolution

Presentation transcript:

Calculus Notes Ch 6.2 Volumes by slicing can be found by adding up each slice of the solid as the thickness of the slices gets smaller and smaller, in other words the number of slices goes to infinity. To find the volume of each slice of the pyramid you would find the area of each square then multiply the area by the thickness of the slice. The thickness would be dx because we are slicing with respect to the x-axis. Next you would add the volumes of each slice together to find the total volume. This would be a Riemann sum with the limit as n the number of slices going to infinity.

Remember the from Ch 5.5 Only now instead of f(x*) we will be summing the areas of the base of each slice. So change f(x*)in the Riemann Sum to A(x*)which represents the area of the base.

Here is an example using a cylinder Find the volume of the cylinder using the formula and slicing with respect to the x-axis. A =  r 2 A =  2 2 = 4 

Now use the formula you learned in Geometry to find the area V=  r 2 h=  (2 2 )(4) = 16  You can also work problems like these with respect to the y-axis.

Volume of solids of revolution using disks If you take the area under the line y = x from 0 to 4 it will look like the diagram below 4 If you rotate this area around the x-axis it will form a cone (see below diagram ) 4

Now use the formula below to find the volume of the 3-D figure formed by rotating around the x-axis. This method is called disks when revolved around the x-axis ( note: it is sliced with respect to the x-axis and is revolved around the x-axis)

Formula for finding the volume of the solid formed when f(x) is revolved around the x- axis Each slice is a circle, the formula for the area of a circle is A=  r 2. The radius is the y-value of the function AKA f(x). So, the area formula becomes A=  (f(x)) 2 Substitute into the formula

Equation y = x from 0 to 4 With geometry V = 1/3  (4 2 )(4)=64  /3 4 With Calculus

Your turn p. 410 # 6. find the volume of the solid that results when the region enclosed by the given curves is revolved around the x-axis y = sec x, x =, x =, y = 0 Use the formula