7.1 Area of a Region Between Two Curves.

Slides:

Advertisements

Similar presentations

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

Advertisements

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

Functions Imagine functions are like the dye you use to color eggs. The white egg (x) is put in the function blue dye B(x) and the result is a blue egg.

6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

APPLICATIONS OF INTEGRATION

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.

Drill Find the area between the x-axis and the graph of the function over the given interval: y = sinx over [0, π] y = 4x-x 3 over [0,3]

7.2 Areas Between Curves. Area Region R is bounded by the curves y = 2 – x 2 and y = -x. Sketch region R. R What is the area of region R?

Area Between Two Curves

In this section, we will investigate the process for finding the area between two curves and also the length of a given curve.

CHAPTER Continuity Areas Between Curves The area A of the region bounded by the curves y = f(x), y = g(x), and the lines x = a, x = b, where f and.

Area Between Two Curves 7.1. Area Formula If f and g are continuous functions on the interval [a, b], and if f(x) > g(x) for all x in [a, b], then the.

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.

Section 5.3: Finding the Total Area Shaded Area = ab y = f(x) x y y = g(x) ab y x Shaded Area =

Area Between Two Curves

APPLICATIONS OF INTEGRATION 6. A= Area between f and g Summary In general If.

7 Applications of Integration

Section 7.2a Area between curves.

Section 5.4 Theorems About Definite Integrals. Properties of Limits of Integration If a, b, and c are any numbers and f is a continuous function, then.

6.1 Areas Between Curves 1 Dr. Erickson. 6.1 Areas Between Curves2 How can we find the area between these two curves? We could split the area into several.

7.2 Areas in the Plane.

1 Solve each: 1. 5x – 7 > 8x |x – 5| < 2 3. x 2 – 9 > 0 :

AREAS USING INTEGRATION. We shall use the result that the area, A, bounded by a curve, y = f(x), the x axis and the lines x = a, and x = b, is given by:

P roblem of the Day - Calculator Let f be the function given by f(x) = 3e 3x and let g be the function given by g(x) = 6x 3. At what value of x do the.

Application: Area under and between Curve(s) Volume Generated

AIM: RANGE OF FUNCTIONS HW P. 26 #52, 64, 66, 101, 103 Do Now: Find the domain of the function and write answer in interval notation. AND.

7.5 Inverses of Functions 7.5 Inverses of Functions Objectives: Find the inverse of a relation or function Determine whether the inverse of a function.

Areas Between Curves In this section we are going to look at finding the area between two curves. There are actually two cases that we are going to be.

APPLICATIONS OF INTEGRATION AREA BETWEEN 2 CURVES.

6.1 Areas Between Curves In this section we learn about: Using integrals to find areas of regions that lie between the graphs of two functions. APPLICATIONS.

Area Between Curves. Objective To find the area of a region between two curves using integration.

Area of R = [g(x) – f(x)] dx

Area of a Region Between Two Curves (7.1)

Chapter 19 Area of a Region Between Two Curves.

Area Between the Curves

7 Applications of Integration

Area of a Region Between 2 Curves

Area Between Two Curves

Section 5.1 Area Between Curves.

Review Calculus.

Area Bounded by Curves sin x cos x.

Lesson 20 Area Between Two Curves

Finding the Area Between Curves

Area of a Region Between Two Curves (7.1)

Section 1.5 Inverse Functions

Lesson 6-1b Area Between Curves.

Find the area of the shaded region

Area & Volume Chapter 6.1 & 6.2 February 20, 2007.

7.1 Area of a Region Between Two Curves

Sec 6.1: AREAS BETWEEN CURVES

SYMMETRY Even and Odd Suppose f is continuous on [-a, a] and even

1.2 Analyzing Graphs of Functions and Relations

Area Between Two Curves

Finding the Total Area y= 2x +1 y= x Area = Area =

Inverse Functions Inverse Functions.

Replacing f(x) with f(x)+k and f(x+k) (2.6.1)

Section 5.3: Finding the Total Area

Linear and Quadratic Functions

Finding the Total Area y= 2x +1 y= x Area = Area =

Analyzing Graphs of Functions

Review 6.1, 6.2, 6.4.

7.1 Area of a Region Between Two Curves.

Warm Up Determine the domain of f(g(x)). f(x) = g(x) =

Ch. 8 – Applications of Definite Integrals

Intersection Method of Solution

(Finding area using integration)

Presentation transcript:

7.1 Area of a Region Between Two Curves

- - = f f f g g g Area of region between f and g Area of region under f(x) = Area of region under g(x)

Ex. Find the area of the region bounded by the graphs of f(x) = x2 + 2, g(x) = -x, x = 0, and x = 1 . Area = Top curve – bottom curve f(x) = x2 + 2 g(x) = -x

Find the area of the region bounded by the graphs of f(x) = 2 – x2 and g(x) = x First, set f(x) = g(x) to find their points of intersection. 2 – x2 = x 0 = x2 + x - 2 0 = (x + 2)(x – 1) x = -2 and x = 1 fnInt(2 – x2 – x, x, -2, 1)

Find the area of the region between the graphs of f(x) = 3x3 – x2 – 10x and g(x) = -x2 + 2x Again, set f(x) = g(x) to find their points of intersection. 3x3 – x2 – 10x = -x2 + 2x 3x3 – 12x = 0 3x(x2 – 4) = 0 x = 0 , -2 , 2 Note that the two graphs switch at the origin.

Now, set up the two integrals and solve.

Find the area of the region bounded by the graphs of x = 3 – y2 and y = x - 1 Area = Right - Left