What type of triangle of given constant perimeter

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What type of triangle of given constant perimeter would provide maximum area?
MATH23 MULTIVARIABLE CALCULUS MULTIPLE INTEGRATION
GENERAL OBJECTIVE Determine the geometric interpretation of partial derivatives and its derivation At the end of the lesson the students are expected to:
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 badxyxf),(dcdyyxf),(Integration with x as the variable of integration with the domain a < x < b 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) Triple Integral: 3! Possible orders of integration: f(x,y,z)  baxgxgdydxyxf)()(21),( dcyhyhdxdyyxf)()(21),( baxyxyyxzyxzdzdydxzyxf)()(),(),(2121),,( fezyzyzyxzyxdxdydzzyxf)()(),(),(2121),,( dcyxyxyxzyxzdzdxdyzyxf)()(),(),(2121),,( baxzxzzxyzxydydzdxzyxf)()(),(),(2121),,( fezxzxzxyzxydydxdzzyxf)()(),(),(2121),,( dcyzyzzyxzyxdxdzdyzyxf)()(),(),(2121),,(
Example: Evaluate 1. 2. 3. 4. 5. .cos320102dxdyxyr drd0sec04.3x28y3dy dxx2x12.
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 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  baxgxgdydxA)()(21 dcyhyhdxdyA)()(21
Example Set up the double integral that gives the area between y = x2and y = x3. Find the area of the region bounded by x = y2and 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 a b c d x y Domain a < x < b c < y < d
Example RdAxya)240()(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. y = g2(x) y = g1(x)  baxgxgdydxyxf)()(21),( dcyhyhdxdyyxf)()(21),(x = h2(x) x = h1(x) (b,d) (a,c)
Example Find the double integral of f(x,y) = 6x2+ 2y over the region enclosed by y = x2and y = 4. Evaluate the integral of  20332yxdxdye
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 + y2And bounded by y = 2x and y = x2.
Bounded by two surfaces The volume bounded by two surfaces can be acquired as:  baxyxylowerupperdydxzzV)()(21)(Where the limits of integration are obtained from the cylinder that contains intersection between the surfaces as projected against the xy-plane.

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