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

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.

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.

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]

AP Calculus Ms. Battaglia

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

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.

Section 3.2 Polynomial Functions and Their Graphs.

Area Between Two Curves

Section 3.2 Notes Writing the equation of a function given the transformations to a parent function.

Section 2.2 Notes: Linear Relations and Functions.

Notes Over 9.7 Using the Discriminant The discriminant is the expression under the radical: If it is Positive: Then there are Two Solutions If it is Zero:

Section 7.2a Area between curves.

5.6 Notes: Find Rational Zeros. Rational Zeros: Where the graph crosses the x-axis at a rational number Rational Zero Theorem: To find the possible rational.

Individual Project ~ FRQ Presentation 2002 AP Calculus AB #1 Marija Jevtic.

Accelerated Math II Polynomial Review. Quick Practice “Quiz” 1. A rectangular sheet of metal 36 inches wide is to be made into a trough by turning up.

Today in Pre-Calculus Go over homework Notes: (need calculator & book)

Warm Up Foil (3x+7)(x-1) Factors, Roots and Zeros.

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.

Section 5.3 Factoring Quadratic Expressions

7.2 Areas in the Plane.

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.

How to solve an AP Calculus Problem… Jon Madara, Mark Palli, Eric Rakoczy.

Brainstorm how you would find the shaded area below.

4.5 Quadratic Equations Wherever the graph of a function f(x) intersects the x-axis, f(x) = 0. A value of x for which f(x) = 0 is a zero of the function.

Example 1A Solve the equation. Check your answer. (x – 7)(x + 2) = 0

FACTOR to SOLVE 1. X 2 – 4x X 2 – 17x + 52 (x-10)(x + 6) x = 10, -6 (x-4)(x - 13) x = 4,13.

Section 5-4(e) Solving quadratic equations by factoring and graphing.

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

3.3 (1) Zeros of Polynomials Multiplicities, zeros, and factors, Oh my.

Systems of Equations A group of two or more equations is called a system. When asked to SOLVE a system of equations, the goal is to find a single ordered.

Solving Quadratic Equations by Factoring. Martin-Gay, Developmental Mathematics 2 Zero Factor Theorem Quadratic Equations Can be written in the form ax.

Solving Quadratic Equations by Factoring. Zero-Product Property If ab=0, then either a=0, b=0 or both=0 States that if the product of two factors is zero.

Solving Polynomials.

APPLICATIONS OF INTEGRATION AREA BETWEEN 2 CURVES.

6.3 Volumes of Revolution Tues Dec 15 Do Now Find the volume of the solid whose base is the region enclosed by y = x^2 and y = 3, and whose cross sections.

Multiple- Choice, Section I Part A Multiple- Choice, Section I Part B Free- Response, Section II Part A Free- Response, Section II Part B # of Questions.

Unit 2 Lesson #3 Tangent Line Problems

Area Between Two Curves Calculus. Think WAY back…

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

Chapter 19 Area of a Region Between Two Curves.

Area of a Region Between 2 Curves

Polynomial Graphs: Zeroes and Multiplicity

Warmup 3-7(1) For 1-4 below, describe the end behavior of the function. -12x4 + 9x2 - 17x3 + 20x x4 + 38x5 + 29x2 - 12x3 Left: as x -,

Section 5.4 Theorems About Definite Integrals

Solving Equations Graphically

Warm Up Graph the following y= x4 (x + 3)2 (x – 4)5 (x – 3)

Area a b x y When the portion of the graph of y = f(x) is below the x-axis between x = a and x = b then the value of will be negative.

Polynomial Multiplicity

Warm Up Graph the following y= x4 (x + 3)2 (x – 4)5 (x – 3)

Quadratics graphs.

Section 7.1 Day 1-2 Area of a Region Between Two Curves

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

Section 5.3: Finding the Total Area

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

Analyzing Graphs of Functions

7.1 Area of a Region Between Two Curves.

Ch. 8 – Applications of Definite Integrals

7.3.2 IB HL Math Year 1.

6-1 System of Equations (Graphing)

Objective SWBAT solve polynomial equations in factored form.

(Finding area using integration)

Presentation transcript:

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

y= f(x) a b y = g(x) A) Area Between Two Curves in [a, b] y= f(x) a b y = g(x) Area under f(x) = Area under g(x) = Area between f(x) and g(x) y = g(x) y= f(x) a b

B) Area Between a Curve and the x-Axis in [a, b] x-Axis is same as y = 0 y = 0 y = g(x) a b y x y = f(x) y x b a Area under f(x) in [a, b] : Top Function is y = f(x) Bottom Function is y = 0 Area under g(x) in [a, b] : Bottom Function is y = g(x) Top Function is y = 0

y = 2x - 1 1) Graph both functions 2) Find the points of intersection by equating both functions: y = y x = -1, x = +3 3) Area Example: Find the area between the graph y= x and y = 2x - 1 y = x C) Area Between Intersecting Curves x = 2x - 1 x 2 - 2x - 3 = 0 (x + 1)(x - 3) = 0 a b y x Top Function is 2x - 1, Bottom Function is x 2 - 4

y x Example: Find the area between the graph y = x 2 - 4x + 3 and the x-axis 1) Graph the function with the x-axis (or y = 0) 2) Find the points of intersection by equating both functions: y = y x = 1, x = 3 y = x 2 - 4x +3 x 2 - 4x + 3 = 0 (x - 1)(x - 3) = ) Area Top Function is y = 0, Bottom Function is y = x 2 - 4x + 3 y = 0

y x Example: Find the area enclosed by the x-axis, y = x 2 - 4, x = -1 and x = 3 D) Area Between Curves With Multiple Points of Intersections (Crossing Curves) 1) Graph the function y = x with the x-axis (or y = 0), Shade in the region between x = -1 and x = 3 2) Find the points of intersection by equating both functions: y = y x = 2, x = -2 x = 0 (x - 2)(x + 2) = 0 3) Area y = x and the x-axis cross each others x = -2, x = 2 y= x 2 – y = 0 or = = A 1 = 9 A 2 = 2.33

y = x y = x 3 Example: Find the area enclosed by y = x 3 and y = x 1) Graph the function y = x 3, and y = x 2) Find the points of intersection by equating both functions: y = y x = 0, x = -1, x = 1 x 3 = x 3) Area The graphs cross at x = -1, x = 0 and x = or = = 0.5 A 1 = 0.25 A 2 = 0.25 x 3 - x = 0 x(x 2 - 1) = 0 x(x - 1)(x + 1) = 0 y x Note: if you write Area =, the answer will be zero.