Use partial integration with respect to y to compute the inner integral (treating x as a constant.)

Transcription

1 Math 54 ~ Multiple Integration 4. Iterated Integrals and Area in the Plane Iterated Integrals f ( x, y) dydx = f ( x, y) dy dx b g ( x) b g ( x) a g ( x) a g ( x) Use partial integration with respect to y to compute the inner integral (treating x as a constant.) f ( x, y) dxdy = f ( x, y) dx dy d h ( y) d h ( y) c h ( y) c h ( y) Use partial integration with respect to x to compute the inner integral (treating y as a constant.) Note: In each of these iterated integrals, the inside limits of integration can be variable with respect to the outer variable of integration. However, the outside limits of integration must be constant with respect to both variables of integration. Definition A vertically simple region,, is a region in the xy-plane that lies between the graphs of two continuous functions of x, that is, = ( x, y) a x b, g ( x) y g ( x) where and g are continuous on [ ab, ]. g Definition A horizontally simple region,, is a region in the xy-plane that lies between the graphs of two continuous functions of y, that is, = ( x, y) c y d, h ( y) x h ( y) where and h are continuous on [ cd, ]. h

2 Area of a egion in the Plane If is defined by a x b and ( y g (, where g and g are continuous on [ ab,, ] then the area of is given by A= b g ( x) dydx a g ( x) If is defined by c y d and ( x h, where h and h are continuous on [, cd ], then the area of is given by A= d h ( y) dxdy c h ( y) Note: If a region is both vertically simple and horizontally simple, then the iterated integral can be ordered in either way, that is, dxdy or dydx. Note: The order of integration can greatly affect the difficulty of the integration! Note: To change the order of integration, it helps to sketch the region.

3 4. Double Integrals and Volume Definition Double Integral If f is defined on a closed, bounded region in the xy-plane, then the double integral of f over is given by n Δ 0 i = f ( xyda, ) = lim f( x, y) ΔA i i i provided the limit exists. If the limit exists, then f is integrable over. Volume of a Solid egion If f is integrable over a plane region and f ( x, y) 0 for all ( x, y ) in, then the volume of the solid region that lies above and below the graph of f is defined as V = f( x, y) da Theorem Properties of Double Integrals Let f and g be continuous over a closed, bounded plane region, and let c be a constant. ) cf( xyda, ) = c f( xyda, ) ) [ f ( xy, ) ± gxyda (, )] = f( xyda, ) ± gxyda (, ) 3) f( x, y) da 0, if f ( x, y) 0 for all ( x, y ) in, if f ( xy, ) gxy (, ) 4) f ( x, y) da g( x, y) da for all ( x, y ) in 5) f ( xyda, ) = f( xyda, ) + f( xyda, ), where is the union of two non- overlapping subregions and.

4 Theorem Fubini s Theorem Let f be continuous on a plane region. ) If is defined by a x b and ( y g (, where g and g are continuous on [ ab],, then b g ( x) f xyda= (, ) f( xydydx, ) a g ( x) ) If is defined by c y d and ( x h (, where h and h are continuous on [ cd],, then d h ( y) f xyda= (, ) f( xydxdy, ) c h ( y)

5 4.3 Change of Variables: Polar Coordinates Definition A polar sector is a region defined by {(, θ ), θ θ θ } r r r r =, where r, r, θ, θ are constants. Definition An r-simple region,, is a region in the rθ -plane that lies between the graphs of two continuous functions of θ, that is, = (, r θ ) α θ β,0 g () θ r g () θ where and g are continuous on [ α, β ]. g Definition An θ -simple region,, is a region in the rθ -plane that lies between the graphs of two continuous functions of r, that is, = (, r θ) r r r, h () r θ h () r where and are continuous on [ r, r ]. h h Theorem Change of Variables to Polar Coordinates Let be a region consisting of all points ( xy, ) = ( rcos θ, rsin θ ) satisfying the conditions 0 g( θ ) r g( θ ), α θ β, where 0 ( β α) π. If g and g are continuous on [ α, β ] and f is continuous on, then β g ( θ ) = f ( x, y) da f ( r cos θ, r sin θ) rdrdθ α g ( θ) Note: There is an extra factor of r in the polar form!!! rdrdθ!!!

6 4.4 Center of Mass and Moments of Inertia Definition of Mass of a Planar Lamina of Variable Density If ρ is a continuous density function on the lamina corresponding to a plane region, then the mass m of the lamina is given by m= ρ( x, y) da Note: If the lamina has a constant density, then ρ ( x, y) is a constant function, call it simply the constant ρ, and then the mass is simply. m= ρda= ρ da= ρa Moments and Center of Mass of a Variable Density Planar Lamina Let ρ be a continuous density function on the planar lamina. The moment of mass with respect to the x- and y-axes are M = yρ( x, y) da and M = xρ( x, y) da. x y If m is the mass of the lamina, then the center of mass is M y M x ( x, y) =, m m. If represents a simple plane region rather than a lamina, the point (, ) x y is called the centroid of the region.

7 4.5 Surface Area Definition of Surface Area If f and its first partial derivatives are continuous on the closed region in the xy-plane, then the area of the surface S given by z = f ( x, y) over is given by Surface Area = ds = + [ fx( x, y)] + [ f y( x, y)] da. S 4.6 Triple Integrals and Applications Definition of Triple Integral If f is continuous over a bounded solid region Q, then the triple integral of f over Q is defined as Q n Δ 0 i = f ( xyzdv,, ) = lim f( x, y, z) ΔV i i i i provided the limit exists. The volume of the solid region Q is given by Volume of Q = dv. Q Theorem Evaluation by Iterated Integrals Let f be continuous on a solid region Q defined by a x b, h( x) y h( x), g( x, y) z g( x, y) where h, h, g, g are continuous functions. Then, b h ( x) g ( x, y) f ( xyzdv,, ) = f( xyzdzdydx,, ). Q a h ( x) g ( x, y)

8 4.7 Triple Integrals in Cylindrical and Spherical Coordinates Cylindrical Coordinates x= rcosθ, y = rsinθ, z = z Given Q is a solid region whose projection onto the xy-plane is the region. Suppose that Q= ( xyz,, ) ( xy, ) is in, h( xy, ) z h( xy, ) and {(, θ ) θ θ θ, ( θ) () θ } = r g r g. If f is a continuous function on the solid Q, you can write the triple integral of f over Q as h ( x, y) f ( xyzdv,, ) = f( xyzdz,, ) da Q h ( x, y) where the double integral over is evaluated in polar coordinates. So, if is r-simple, the iterated form of the triple integral in cylindrical form is θ g ( θ) h ( rcos θ, rsin θ) = f ( x, y, z) dv f ( r cos θ, r sin θ, z) rdzdrdθ Q θ g ( θ) h ( rcos θ, rsin θ) Spherical Coordinates x = ρ sinφcosθ, y = ρ sinφsinθ, z = ρ cosφ f ( xyzdv,, ) = f( ρ sinφcos θ, ρsinφsin θ, ρcos φ) ρ sinφdρdφdθ Q θ φ ρ θ φ ρ Note: These formulas can be modified for different orders of integration and generalized to include regions with variable boundaries.

9 4.8 Change of Variables: Jacobians Definition of the Jacobian If x = guv (, ) and y = h( u, v), then the Jacobian of x and y with respect to u and v, denoted by ( x, y) ( u, v), is x x ( x, y) u v x y y x = = ( uv, ) y y u v u v u v Theorem Change of Variables for Double Integrals Let and S be regions in the xy- and uv-planes (respectively) that are related by the equations x = guv (, ) and y = h( u, v) such that each point in is the image of a unique point in S. If f is continuous on, g and h have continuous partial derivatives on S, and ( x, y) ( u, v) is nonzero on S, then ( xy, ) f ( x, y) dxdy = f ( g( u, v), h( u, v)) dudv ( uv, ) S