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

Slides:



Advertisements
Similar presentations
7.2: Volumes by Slicing Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 Little Rock Central High School, Little Rock,
Advertisements

Golden Spike National Historic Site, Promontory, Utah Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, Day 1 Lengths.
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.
The Shell Method Volumes by Cylindrical Shells By Christine Li, Per. 4.
Volume: The Disk Method
Finding the Area and the Volume of portions under curves.
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.
7.2: Volumes by Slicing – Day 2 - Washers Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 Little Rock Central High School,
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.
Integral Calculus One Mark Questions. Choose the Correct Answer 1. The value of is (a) (b) (c) 0(d)  2. The value of is (a) (b) 0 (c) (d) 
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.
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, B Volumes by the Washer Method Limerick Nuclear Generating Station,
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,
Applications of Integration Copyright © Cengage Learning. All rights reserved.
Do Now: #10 on p.391 Cross section width: Cross section area: Volume:
Golden Spike National Historic Site, Promontory, Utah Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, Day 1 Lengths.
7.4 Lengths of Curves. 2+x csc x 1 0 If we want to approximate the length of a curve, over a short distance we could measure a straight line. By the.
Chapter 6 – Applications of Integration 6.3 Volumes by Cylindrical Shells 1Erickson.
Copyright © Cengage Learning. All rights reserved. 8 Further Applications of Integration.
Volumes of Revolution Disks and Washers
Chapter 5: Integration and Its Applications
Arc Length and Surfaces of Revolution
7.4 Length of a Plane Curve y=f(x) is a smooth curve on [a, b] if f ’ is continuous on [a, b].
Volume: The Disk Method
Volumes of Revolution The Shell Method Lesson 7.3.
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: 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
7.4 Day 2 Surface Area Greg Kelly, Hanford High School, Richland, Washington(Photo not taken by Vickie Kelly)
Applications of Integration Copyright © Cengage Learning. All rights reserved.
8.2 Area of a Surface of Revolution
10.3 day 2 Calculus of Polar Curves Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 Lady Bird Johnson Grove, Redwood National.
Volumes Lesson 6.2.
Chapter 8 – Further Applications of Integration
Do Now: #16 on p.518 Find the length of the curve. Evaluate numerically…
Golden Spike National Historic Site, Promontory, Utah Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, Day 1 Lengths.
7.4 Day 2 Surface Area Greg Kelly, Hanford High School, Richland, Washington(Photo not taken by Vickie Kelly)
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 9.2 Area of a Surface of Revolution. THE AREA OF A FRUSTUM The area of the frustum of a cone is given by.
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 =
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.
Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College Photo by Vickie Kelly, Day 3 The Shell Method.
10.3 Parametric Arc Length & Area of a Surface of Revolution.
Copyright © Cengage Learning. All rights reserved. 8.2 Area of a Surface of Revolution.
Volume: The Shell Method 7.3 Copyright © Cengage Learning. All rights reserved.
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.
5053 -Volume by Shells AP Calculus.
Solids of Revolution Shell Method
Solids of Revolution Shell Method
1.) Set up the integral to find the area of the shaded region
10.6: The Calculus of Polar Curves
7.4 Lengths of Curves and Surface Area
Arc Length and Surfaces of Revolution
Double Integration Greg Kelly, Hanford High School, Richland, Washington.
Volumes – The Disk Method
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.
7.4 Arc Length and Surface of Revolution
7.4 Lengths of Curves and Surface Area
Review 6.1, 6.2, 6.4.
7.4 Day 2 Surface Area (Photo not taken by Vickie Kelly)
Area of a Surface of Revolution
6.4 Arc Length and Surface of Revolution
Presentation transcript:

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

Surface Area: r Consider a curve rotated about the x -axis: The surface area of this band is: The radius is the y -value of the function, so the whole area is given by: This is the same ds that we had in the “length of curve” formula, so the formula becomes: Surface Area about x -axis (Cartesian): To rotate about the y -axis, the radius is just x in the formula.

Surface Area: r Surface Area about y -axis (Cartesian): Consider a curve rotated about the y -axis: The surface area of this band is: The radius is the x -value of the function, so the whole area is given by: This is the same ds that we had in the “length of curve” formula, so the formula becomes: Not needed for rotating about y-axis.

Surface Area: r Difference between surface area and shell method: With surface area, we are integrating with respect to the change is s (ds) so summing up the arc length. We find volumes with shell method because we are integrating the change in x (dx) so accumulating the thicknesses to find the volume.

Example: Rotate about the y -axis.

Example: Rotate about the y -axis.

Example: Rotate about the y -axis. From geometry:

Example: rotated about x -axis. The TI-89 gets:

Example: The Area of a Surface of Revolution Find the area of the surface formed by revolving the graph of f(x) = x 3 on the interval [0, 1] about the x-axis, as shown in Figure Figure 7.46

Solution: The distance between the x-axis and the graph of f is r(x) = f(x), and because f'(x) = 3x 2, the surface area is

Example: The Area of a Surface of Revolution Find the area of the surface formed by revolving the graph of f(x) = x 2 on the interval [0, ] about the y-axis, as shown.

Solution: The Area of a Surface of Revolution In this case, the distance between the graph of f and the y-axis is Using the surface area is:

Homework: 7.4 day 2: MMM pgs. 59 & 62.