MULTIPLE INTEGRATION MATH23 MULTIVARIABLE CALCULUS GENERAL OBJECTIVE At the end of the lesson the students are expected to: • Determine the geometric interpretation of partial derivatives and its derivation 14.1.1 (p. 1001) The Volume Problem Figure 14.1.2 Figure 14.1.3 Definition 14.1.2 (p. 1002) Definition: Partial Definite Integrals b a f ( x, y)dx Integration with x as the variable of integration with the domain a<x<b d c f ( x, y)dy Integration with y as the variable of integration with the domain c<y<d • Evaluation of Multiple Integrals: Double Integral: 2! Possible orders of integration: f(x,y) b g2 ( x) a g1 ( x ) d h2 ( y ) c h1 ( y ) f ( x, y )dydx f ( x, y )dxdy Triple Integral: 3! Possible orders of integration: f(x,y,z) b y2 ( x ) a y1 ( x ) d x2 ( y ) c x1 ( y ) f y2 ( z ) e y1 ( z ) z2 ( x , y ) z2 ( x , y ) x2 ( y , z ) z1 ( x , y ) z1 ( x , y ) x1 ( y , z ) f ( x, y, z )dzdydx f ( x, y, z )dzdxdy f ( x, y, z )dxdydz b z2 ( x ) y2 ( x , z ) a z1 ( x ) d z2 ( y ) x2 ( y , z ) c z1 ( y ) f x2 ( z ) y2 ( x , z ) x1 ( z ) y1 ( x , z ) e y1 ( x , z ) x1 ( y , z ) f ( x, y, z )dydzdx f ( x, y, z )dxdzdy f ( x, y, z )dydxdz Example: Evaluate 2 1 1. 2 3 y cos xdydx. 0 0 2. 4 sec r dr d . 0 0 2 2x 3. 2 3 3x 8y dy dx . 1 x 4. 5. Applications of Multiple Integration • Areas by double integration • Volume by double Integration – Rectangular Base – Base bounded by given Curves – Solids bounded by two surfaces • Volume by triple integration Area by Double Integral If a region R is bounded below by y = g1(x) and above by y = g2(x), and by a < x < b, then the area is given by A b a g2 ( x) g1 ( x ) dydx Consequently, if a region R is bounded on the left x= h1(y) and to the right by x = h2(y), and by c < y < d, then the area is given by A d c h2 ( y ) h1 ( y ) dxdy Example Set up the double integral that gives the area between y = x2 and y = x3. Find the area of the region bounded by x = y2 and y = x. Double Integral for Volumes Let R be a region in the xy-plane and T be the solid bounded below by R and bounded above by the surface z = f(x,y). Then the volume of T is found by V= ∫∫ f(x,y) dxdy Evaluation of Multiple Integral (Double Integral) A. Rectangular Based Solids y d c a b x • Domain a<x<b c<y<d Example (a) (40 2 xy)dA R Where R is the rectangle: 1≤ x ≤ 3 ; 2 ≤ y ≤ 4 1. Determine the volume above the xy plane and below the Surface: z = 5 – x2 - y 2 and bounded by the Domain -1 < x < 1 0<y<1 Volumes by Multiple Integration • Solids Bounded by curves at the base. (b,d) y = g2 (x) x = h1 (x) x = h2 (x) y = g1 (x) (a,c) b g2 ( x) a g1 ( x ) f ( x, y )dydx d h2 ( y ) c h1 ( y ) f ( x, y )dxdy Example Find the double integral of f(x,y) = 6x2 + 2y over the region enclosed by y = x2 and y = 4. Evaluate the integral of 2 3 0 3y x2 e dxdy Example: Double Integral for Volumes Set up the integral to find the volume of the solid that lies below the cone z = 4 – (x2 + y2)1/2 and above the xy-plane Example Setup and determine the volume above the xy – plane and below the paraboloid z = x2 + y2 And bounded by y = 2x and y = x2. Bounded by two surfaces • The volume bounded by two surfaces can be acquired as: V b a y2 ( x ) y1 ( x ) ( zupper zlower ) dydx • Where the limits of integration are obtained from the cylinder that contains intersection between the surfaces as projected against the xy-plane. Example: Bounded by two surfaces Set up the double integral that gives the volume of the solid that lies below the upper part of the sphere x2 + y2 + z2 = 6 and above the paraboloid z = x2 + y2 . DO NOT evaluate the integral. Example: Set up the integral of f(x,y,z) over G, the solid “ice cream cone” bounded by the cone z = (x2 + y2)1/2 and the sphere z = (1 – x2 – y2)1/2. Definition: Triple Integral Let f(x,y,z) be the density of some 3-dimensional solid W. Objective: Define the triple integral of f over W. Example: Set-up the integral to find the volume of the solid enclosed between the paraboloids z = 5x2 + 5y2 and z = 6 – 7x2 – y2 Example: Definition: Riemann Sum Value Definition Example: Set up the integral of f(x,y,z) over G, the solid “ice cream cone” bounded by the cone z = (x2 + y2)1/2 and the sphere z = (1 – x2 – y2)1/2. Example: Assignment:!!!!!!!!!!!!!!!!! Use triple integral to find the volume of the solid within the cylinder x2 + y2 = 9 and between the plane z = 1 and x + z = 5. Example: Set-up the integral to find the volume of the solid enclosed between the paraboloids z = 5x2 + 5y2 and z = 6 – 7x2 – y2 Example:
© Copyright 2024