Mathematics.

Slides:



Advertisements
Similar presentations
- 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.
Advertisements

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.
Applications of Integration Volumes of Revolution Many thanks to od/gallery/gallery.html.
Applications Of The Definite Integral The Area under the curve of a function The area between two curves The Volume of the Solid of revolution.
MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.1 – Area Between Two Curves Copyright © 2006 by Ron Wallace, all.
Applications Of The Definite Integral The Area under the curve of a function The area between two curves.
7.1 Area of a Region Between Two Curves.
Warm Up Show all definite integrals!!!!! 1)Calculator Active: Let R be the region bounded by the graph of y = ln x and the line y = x – 2. Find the area.
A REA B ETWEEN THE C URVES 4-D. If an area is bounded above by f(x) and below by g(x) at all points on the interval [a,b] then the area is given by.
Do Now: #10 on p.391 Cross section width: Cross section area: Volume:
4.6 Area Between Curves We applied the notion of the integral to calculate areas of only one type: the area under a curve bounded by the x-axis. Now, we.
Section 7.2a Area between curves.
Section 5.3: Evaluating Definite Integrals Practice HW from Stewart Textbook (not to hand in) p. 374 # 1-27 odd, odd.
Mathematics. Session Three Dimensional Geometry–1(Straight Line)
The Fundamental Theorem of Calculus Lesson Definite Integral Recall that the definite integral was defined as But … finding the limit is not often.
AP Calculus Definite Integrals Review (sections )
Volumes of Revolution Disks and Washers
Section 15.3 Area and Definite Integral
In this section, we will introduce the definite integral and begin looking at what it represents and how to calculate its value.
Mathematics. Session Definite Integrals –1 Session Objectives  Fundamental Theorem of Integral Calculus  Evaluation of Definite Integrals by Substitution.
SPECIALIST MATHS Calculus Week 6 Definite Integrals & Areas.
4-3: Riemann Sums & Definite Integrals
Definite Integrals Riemann Sums and Trapezoidal Rule.
Warm Up 1) 2) 3)Approximate the area under the curve for 0 < t < 40, given by the data, above using a lower Reimann sum with 4 equal subintervals. 4)Approximate.
STROUD Worked examples and exercises are in the text PROGRAMME F12 INTEGRATION.
19/06/2007VER2-0 (a) Find the Area bounded by the curve y=20sin200  t and the t axis. Between t=0 and t=30ms EXERCISE SET 6 – Area under a Curve & Mean.
Mathematics.
4.2 Area Definition of Sigma Notation = 14.
4.3 Riemann Sums and Definite Integrals Definition of the Definite Integral If f is defined on the closed interval [a, b] and the limit of a Riemann sum.
Definite Integral df. f continuous function on [a,b]. Divide [a,b] into n equal subintervals of width Let be a sample point. Then the definite integral.
5.3 Definite Integrals and Riemann Sums. I. Rules for Definite Integrals.
Applications of Integration 7 Copyright © Cengage Learning. All rights reserved.
5.1 Areas and Distances. Area Estimation How can we estimate the area bounded by the curve y = x 2, the lines x = 1 and x = 3, and the x -axis? Let’s.
DO NOW: v(t) = e sint cost, 0 ≤t≤2∏ (a) Determine when the particle is moving to the right, to the left, and stopped. (b) Find the particles displacement.
(MTH 250) Lecture 19 Calculus. Previous Lecture’s Summary Definite integrals Fundamental theorem of calculus Mean value theorem for integrals Fundamental.
FINDING VOLUME USING DISK METHOD & WASHER METHOD
TOPIC : 7 NUMERICAL METHOD.
7 Applications of Integration
6.6 Area Between Two Curves
The Area Between Two Curves
Area Between the Curves
7 Applications of Integration
Area of a Region Between 2 Curves
Section 5.1 Area Between Curves.
Integration Review Problems
The Area Question and the Integral
Lesson 20 Area Between Two Curves
Finding the Area Between Curves
Volumes of Revolution Disks and Washers
Volumes – The Disk Method
Section 5.4 Theorems About Definite Integrals
The Area Between Two Curves
Mathematics.
Area of a Region Between Two Curves (7.1)
4 Integrals.
5.4 – Applying Definite Integration
Volumes – The Disk Method
7.2 Area Between Two Curves
Mathematics.
Applications of Integration
The Fundamental Theorem of Calculus
7 Applications of Integration
4.4 The Fundamental Theorem of Calculus
Objectives Approximate a definite integral using the Trapezoidal Rule.
7.1 Area of a Region Between Two Curves.
Jay James Mike Norton Stephen Nesemann Kaitlyn Richardson
Symbolic Integral Notation
7 Applications of Integration
7.3.2 IB HL Math Year 1.
(Finding area using integration)
Presentation transcript:

Mathematics

Session Definite Integrals - 3

Session Objectives Definite Integral as the Limit of a Sum Areas of Bounded Regions Class Exercise

Definite Integral as the Limit of a Sum OR

Example - 1

Solution Cont.

Example - 2

Solution Cont.

Example - 3

Solution Cont.

Example - 4

Solution Cont.

Example - 5

Solution Cont.

Areas of Bounded Regions 1. Let f(x) be a continuous function defined on the interval [a, b]. Then, the area bounded by the curve y = f(x), x-axis and the ordinates x = a, x = b is The area bounded by the curve x = f(y), y-axis and the abscissae y = c, y = d is

Areas of Bounded Regions Cont.

Example - 6

Solution Cont.

Example - 7

Solution Cont.

Solution Cont.

Example - 8 Solution: The given curves are (i) and (ii) intersect at (1, 0) and (0, 1).

Solution Cont.

Example - 9

Solution Cont.

Example - 10 Solution: The given curves are

Solution Cont.

Solution Cont.

Solution Cont.

Example - 11

Solution Cont. x' x y y' O (1, 0) x2 + y2 = 1 (x - 1)2 + y2 = 1

Solution Cont.

Solution Cont.

Thank you