INTEGRATION APPLICATIONS 1

Slides:

Advertisements

Similar presentations

Higher Unit 2 Outcome 2 What is Integration

Advertisements

“Teach A Level Maths” Vol. 1: AS Core Modules

Integration It is sometimes possible to simplify an integral by changing the variable. This is known as integration by substitution. Use the substitution:

Do Now: p.381: #8 Integrate the two parts separately: Shaded Area =

Definite Integration and Areas 01 It can be used to find an area bounded, in part, by a curve e.g. gives the area shaded on the graph The limits of integration...

5.4 The Fundamental Theorem. The Fundamental Theorem of Calculus, Part 1 If f is continuous on, then the function has a derivative at every point in,

Section 7.2a Area between curves.

Section 15.3 Area and Definite Integral

Section 5.2: Definite Integrals

Areas & Definite Integrals TS: Explicitly assessing information and drawing conclusions.

Introduction This chapter focuses on Parametric equations Parametric equations split a ‘Cartesian’ equation into an x and y ‘component’ They are used.

Introduction We have seen how to Integrate in C1 In C2 we start to use Integration, to work out areas below curves It is increasingly important in this.

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.

Pre-Algebra 2-2 Subtracting Integers 2-2 Subtracting Integers Pre-Algebra Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson.

EXAMPLE 1 Subtracting Integers –56 – (–9) = – = –47 –14 – 21 = –14 + (–21) = –35 Add the opposite of –9. Add. Add the opposite of 21. Add. a. –56.

Definite Integration and Areas 01 It can be used to find an area bounded, in part, by a curve e.g. gives the area shaded on the graph The limits of integration...

STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 INTEGRATION APPLICATIONS 1 PROGRAMME 19.

Chapter 7 – Techniques of Integration 7.3 Trigonometric Substitution 1Erickson.

Integration – Overall Objectives  Integration as the inverse of differentiation  Definite and indefinite integrals  Area under the curve.

Applications of Integration 6. More About Areas 6.1.

5.3 Definite Integrals. Example: Find the area under the curve from x = 1 to x = 2. The best we can do as of now is approximate with rectangles.

VARIABLES AND EXPRESSIONS. Definitions: Algebra – Is a language of symbols, including variables. Variable - Is a letter or symbol that represents a quantity.

Copyright © Cengage Learning. All rights reserved. 6 Applications of Integration.

[5-4] Riemann Sums and the Definition of Definite Integral Yiwei Gong Cathy Shin.

WBHS Advanced Programme Mathematics

Subtracting Integers #41.

Definite integration results in a value.

Copyright © Cengage Learning. All rights reserved.

2.8 Integration of Trigonometric Functions

Copyright © Cengage Learning. All rights reserved.

The Area Between Two Curves

Equations Quadratic in form factorable equations

8-2 Areas in the plane.

2-2 Subtracting Integers Warm Up Problem of the Day

Anti-differentiation

PROGRAMME F12 INTEGRATION.

Integration Finding the Area Under a Curve & the Area Between Two Lines AS Maths with Liz.

Integration Review Problems

Area Bounded by Curves sin x cos x.

Finding the Area Between Curves

PROGRAMME 23 MULTIPLE INTEGRALS.

Which of the equations below is an equation of a cone?

C2 – Integration – Chapter 11

Always remember to add the constant for this type of question

Area of a Region Between Two Curves (7.1)

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Coordinate Geometry in the (x,y) plane.

Evaluate the integral. {image}

Evaluate the integral. {image}

Parametric equations Problem solving.

Challenging problems Area between curves.

Evaluating Expressions

PROGRAMME F13 INTEGRATION.

Definite Integration.

PROGRAMME 17 INTEGRATION 2.

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Definition: Sec 5.2: THE DEFINITE INTEGRAL

Equations Quadratic in form factorable equations

Section Volumes by Slicing

More Definite Integrals

6-2 definite integrals.

(Finding area using integration)

Evaluating an expression with one variable

Presentation transcript:

INTEGRATION APPLICATIONS 1 PROGRAMME 18 INTEGRATION APPLICATIONS 1

Programme 18: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

Programme 18: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

Programme 18: Integration applications 1 Basic applications Areas under curves Definite integrals

Programme 18: Integration applications 1 Basic applications Areas under curves The area above the x-axis between the values x = a and x = b and beneath the curve in the diagram is given as the value of the integral evaluated between the limits x = a and x = b: where

Programme 18: Integration applications 1 Basic applications Areas under curves If the integral is negative then the area lies below the x-axis. For example:

Programme 18: Integration applications 1 Basic applications Definite integrals The integral with limits is called a definite integral:

Programme 18: Integration applications 1 Basic applications Definite integrals To evaluate a definite integral: Integrate the function (omitting the constant of integration) and enclose within square brackets with the limits at the right-hand end. Substitute the upper limit. Substitute the lower limit Subtract the second result from the first result.

Programme 18: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

Programme 18: Integration applications 1 Parametric equations If a curve has parametric equations then: Express x and y in terms of the parameter Change the variable Insert the limits of the parameter

Programme 18: Integration applications 1 Parametric equations For example, if x = a sin and y = b cos then the area under the curve y between  = 0 and  =  is:

Programme 18: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

Programme 18: Integration applications 1 Mean values The mean value M of a variable y = f (x) between the values x = a and x = b is the height of the rectangle with base b – a and which has the same area as the area under the curve:

Programme 18: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

Programme 18: Integration applications 1 Root mean square (rms) values The root mean square value of y is the square root of the mean value of the squares of y between some stated limits:

Programme 18: Integration applications 1 Learning outcomes Evaluate the area beneath a curve Evaluate the area beneath a curve given in parametric form Determine the mean value of a function between two points Evaluate the root mean square (rms) value of a function