Do Now: #16 on p.518 Find the length of the curve. Evaluate numerically…

Slides:

Advertisements

Similar presentations

DO NOW: Find the volume of the solid generated when the

Advertisements

 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

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.

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

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

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.

 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.

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.

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

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) 

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.

Section 6.4 Use point plotting to graph polar equations.

Section 1.1 The Distance and Midpoint Formulas. x axis y axis origin Rectangular or Cartesian Coordinate System.

The Coordinate Plane coordinate plane In coordinate Geometry, grid paper is used to locate points. The plane of the grid is called the coordinate plane.

Applications of Integration Copyright © Cengage Learning. All rights reserved.

Midpoint formula: Distance formula: (x 1, y 1 ) (x 2, y 2 ) 1)(- 3, 2) and (7, - 8) 2)(2, 5) and (4, 10) 1)(1, 2) and (4, 6) 2)(-2, -5) and (3, 7) COORDINATE.

 We can define both elements of the ordered pair, (x, y), in terms of another variable, t, called a parameter.  Example: Given and, a) Find the points.

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

Chapter 6 Additional Topics in Trigonometry Copyright © 2014, 2010, 2007 Pearson Education, Inc Graphs of Polar Equations.

Chapter 8 Plane Curves and Parametric Equations. Copyright © Houghton Mifflin Company. All rights reserved.8 | 2 Definition of a Plane Curve.

10.2 – 10.3 Parametric Equations. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations.

Section 17.5 Parameterized Surfaces

Vector Functions 10. Parametric Surfaces Parametric Surfaces We have looked at surfaces that are graphs of functions of two variables. Here we.

Conics, Parametric Equations, and Polar Coordinates Copyright © Cengage Learning. All rights reserved.

Section 6.5 Area of a Surface of Revolution. All graphics are attributed to:  Calculus,10/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright.

HPC 1.3 Notes Learning Targets: - Test an equation for symmetry with respect to: x-axis, y-axis, or origin - Know how to graph key equations - Write the.

Volumes of Revolution Disks and Washers

1.3 Symmetry; Graphing Key Equations; Circles

7.4 Length of a Plane Curve y=f(x) is a smooth curve on [a, b] if f ’ is continuous on [a, b].

Parametric Equations. In a rectangular coordinate system, you will recall, a point in the plane is represented by an ordered pair of number (x,y), where.

Vectors and the Geometry of Space Copyright © Cengage Learning. All rights reserved.

Sullivan Algebra and Trigonometry: Section 10.2 Objectives of this Section Graph and Identify Polar Equations by Converting to Rectangular Coordinates.

Solids of Revolution Disk Method

Conics, Parametric Equations, and Polar Coordinates 10 Copyright © Cengage Learning. All rights reserved.

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

Cylinders and Quadratic Surfaces A cylinder is the continuation of a 2-D curve into 3-D.  No longer just a soda can. Ex. Sketch the surface z = x 2.

Lengths and Surface Areas of Polar Curves Section 10.6b.

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

Warm Up Week 2. Section 10.6 Day 1 I will write the equation of a circle. Circle Equation Must know the coordinate of the center and the radius.

Volume: The Shell Method

Chapter 8 – Further Applications of Integration

9.6 Circles in the Coordinate Plane Date: ____________.

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 7.3 Solids of Revolution.

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

Notes Over 1.1 Checking for Symmetry Check for symmetry with respect to both axis and the origin. To check for y-axis symmetry replace x with  x. Sym.

Section Volumes by Slicing 7.3 Solids of Revolution.

LENGTHS OF PARAMETRIZED CURVES Section 10.1a. Definition: Derivative at a Point A parametrized curve,,, has a derivative at if and have derivatives at.

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

STROUD Worked examples and exercises are in the text Programme 20: Integration applications 2 INTEGRATION APPLICATIONS 2 PROGRAMME 20.

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.

10.3 Parametric Arc Length & Area of a Surface of Revolution.

Equation of a Circle. Equation Where the center of the circle is (h, k) and r is the radius.

C.2.5b – Volumes of Revolution – Method of Cylinders Calculus – Santowski 6/12/20161Calculus - Santowski.

Section 9.2: Parametric Equations – Slope, Arc Length, and Surface Area Slope and Tangent Lines: Theorem. 9.4 – If a smooth curve C is given by the equations.

  Where the center of the circle is (h, k) and r is the radius. Equation.

Calculus 6-R Unit 6 Applications of Integration Review Problems.

Volumes of Solids of Rotation: The Disc Method

Homework Questions…. Pre-Calculus Section 6.3 in your book…

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

Parametric Equations and Polar Coordinates

Volumes of Solids of Revolution

Length of Curves and Surface Area

Presentation transcript:

Do Now: #16 on p.518 Find the length of the curve. Evaluate numerically…

Surface Areas of Parametrized Curves Section 10.1b

Surface Area (from a Smooth Parametrized Curve) If a smooth curve,,, is traversed exactly once as increases from to, then the areas of the surfaces generated by revolving the curve about the coordinate axes are as follows. 1. Revolution about the x-axis : 2. Revolution about the y-axis :

The standard parametrization of the circle with radius 1 centered at the point (0, 1 ) in the xy-plane is Use this parametrization to find the area of the surface swept out by revolving the circle about the x-axis (look at Figure 10.4 on p.517).

The standard parametrization of the circle with radius 1 centered at the point (0, 1 ) in the xy-plane is Use this parametrization to find the area of the surface swept out by revolving the circle about the x-axis (look at Figure 10.4 on p.517).

Guided Practice Find the area of the surface generated by revolving the curve about the indicated axis. y-axis Graph the curve and visualize the surface…

Guided Practice Find the area of the surface generated by revolving the curve about the indicated axis. y-axis

Guided Practice Find the area of the surface generated by revolving the curve about the indicated axis. y-axis

Guided Practice Find the area of the surface generated by revolving the curve about the indicated axis. x-axis Graph the curve and visualize the surface…

Guided Practice Find the area of the surface generated by revolving the curve about the indicated axis. x-axis

Guided Practice Find the area of the surface generated by revolving the curve about the indicated axis. x-axis