![]() (Visualize!) HW 13, 14 e.In this unit we will learn about derivatives of functions of several variables. Area of the part of the paraboloid z x2 C y2 that lies under the plane z 9. Area of the portion of the surface z f x, y on Consider r x, y x i C y j C f x, y k. The surface with parametric equations x u2, y uv, e. The part of the plane 2 x K5 y C z 10 that lies above the triangle with vertices (0,0), (0,6), and (4, 0). D The part of the plane x C 2 y C 3 z 1 that lies inside the cylinder x2 C y2 3. Area of the surface given as two variable function Figure 2, Figure 3 Definition : Surface Area If the surface is given as r u, v x u, v i C y u, v j C z u, v k Area (surface) ru rv d A for u, v 2 D. Area of the surface given as parametric function. Volume of the tetrahedron bounded the planes x C 2 y C z 2, x 2 y, x 0, and z 0. Evaluate x y dA, where D is the region bounded the line y x K 1, and the parabola D y2 2 x C 6. Find the volume of the solid that lies under z f x, y x2 C y2 and above the region D which is the region bounded the parabolas y 2 x and y x2. Volume of the solid bounded the elliptic paraboloid x2 C 2 y2 C z 16, the planes x 2, y 2, three coordinate planes. a) 0 2 3 b) 1 x2 y dy dx 1 x2 y dx dy 0 Theorem If f is continuous on the rectangle R b d f x, y dA d b f x, y dy dx a c x, y a x b, c y d then f x, y dx dy c a R SO! If the function is nice enough to be continuous double integral is interated integral and it is easy just like Cal I. Using Geometry x, y K1 x 1, K2 y 2, 1 Kx2 D A R Section 12 Iterated Integrals 3 2 e. Divide R into four equal squares and choose the sample point to be the upper right corner of each square Rij. Use double Riemann sum tp estimate a volume of the solid that lies above the square R 0, 2 0, 2 and below the elliptic paraboloid z 16 Kx2 K2 y2. Figure 3, 4, 5 m f x, y D A R Double Integral lim m, n n f, DA lim Double Riemann sum (Double Integral of f over the rectangle R if the limit exists, and f is called integrable.) e. b n f x dx n lim D x a Figure 1 Cal : Volume under the surface over the rectangle R. Preview text MTH 203 Multivariate Calculus Chapter 12 Multiple Integrals Section 12 Double Integrals over Rectangles Cal I : Area under the curve on the interval (a, b). ![]() Evaluate integral e xCy xKy dA R where R is the trapezoid region with vertices (1,0), Right Transformation? HW 15, 16, 17,18,19,23 ![]()
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