MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 3 Double Integrals 3A Double Integrals in Rectangular Coordinates 3A 1 Evaluate each of the following iterated integrals 6x2 2y dy dx b l n I 2L n sin t t cos U dt du 3A 2 Express each double integral over the given region R as an iterated integral using the given order of integration Use the method described in Notes I to supply the limits of integration For some of them it may be necessary to break the integral up into two parts In each case begin by sketching the region a R is the triangle with vertices at the origin 0 2 and 2 2 Express as an iterated integral i dy dx ii L dx dy b R is the finite region between the parabola y 2x x2 and the x axis Express as an iterated integral i L L L dy dx ii L dx dy c R is the sector of the circle with center at the origin and radius 2 lying between the x axis and the line u x Express as an iterated integral i dy dx ii d R is the finite region lying between the parabola having slope 1 Express as an iterated integral i dy dx L y2 ii dx dy x and the line through 2 O dx dy 3A 3 Evaluate each of the following double integrals over the indicated region R Choose whichever order of integration seems easier this will be influenced both by the integrand and by the shape of R a L x dA R is the finite region bounded by the axes and 2y b L 2 x and c y2 Y2 dA x 2 R is the finite region in the first quadrant bounded by the axes 1 x dx dy is easier L y dA R is the triangle with vertices at f1 O 0 l 3A 4 Find by double integration the volume of the following solids a the solid lying under the graph of z sin2 x and over the region R bounded below by the x axis and above by the central arch of the graph of cosx b the solid lying over the finite region R in the first quadrant between the graphs of x and x2 and underneath the graph of z xy c the finite solid lying underneath the graph of x2 y2 above the xy plane and between the planes x 0 and x 1 E 18 02 EXERCISES 2 3A 5 Evaluate each of the following iterated integrals by changing the order of integration begin by figuring out what the region R is and sketching it a Jd2l2 e Y2dydx b Jdli4 1 dudt dudx 3A 6 Each integral below is over the disc consisting of the interior R of the unit circle centered at the origin For each integral use the symmetries of R and the integrand i to identify its value as zero or if its value is not zero ii to find a double integral which is equivalent i e has the same value but which has a simpler integrand and or is taken over the first quadrant if possible or over a half disc Do not evaluate the integral 3A 7 By using the inequality f 5 g on R estimates are valid dA area of R a 1 4 4 b JJ JJ i J f dA 5 JJR gdA show the following x dA x2 y2 35 R is the square 0 x y 1 3B Double Integrals in Polar Coordinates In evaluating the integrals the following definite integrals will be useful sinnx dx For example 1 3 5 n l n2 4 n 2 2 4 n 1 1 3 n cosnx dx if n is an even integer 2 2 if n is an odd integer 2 3 Jd and the same holds if cosx is substituted for sinx 3B 1 Express each double integral over the given region R as an iterated integral in polar coordinates Use the method described in Notes I to supply the limits of integration For some of them it may be necessary to break the integral up into two parts In each case begin by sketching the region a The region lying to the right of the circle with center at the origin and radius 2 and to the left of the vertical line through 1 O b The circle of radius 1 and center at 0 l c The region inside the cardioid center at the origin T 1 cos8 and outside the circle of radius 312 and d The finite region bounded by the y axis the line y a and a quarter of the circle of radius a and center at a 0 3 DOUBLE INTEGRALS 3 3B 2 Evaluate by iteration the double integrals over the indicated regions Use polar coordinates a C LF SL SL R is the region inside the first quadrant loop of r sin20 R is the first quadrant portion of the interior of x2 y2 a 2 R is the triangle with vertices at 0 O 1 O 1 l tan2 0 dA dx dy R is the right half disk of radius centered at 0 3B 3 Find the volumes of the following domains by integrating in polar coordinates a a solid hemisphere of radius a place it so its base lies over the circle x2 y2 a2 b the domain under the graph of xy and over the quarter disc region R of 3B 2b c the domain lying under the cone z d center at 0 l m and over the circle of radius one and d the domain lying under the paraboloid z x2 right hand loop of r2 cos 0 y2 and over the interior of the 3B 4 Sometimes students wonder if you can do a double integral in polar coordinates iterating in the opposite order J d0 dr Though this is uncommon just to see if you can carry out in a new situation the basic procedure for putting in the limits try supplying the limits for this integral over the region bounded above by the lines x 1 and y 1 and below by a quarter of the circle of radius 1 and center at the origin 3C Applications of Double Integration If no coordinate system is specified for use you can use either rectangular or polar coordinates whichever is easier In some of the problems a good placement of the figure in the coordinate system simplifies the integration a lot 3C 1 Let R be a right triangle with legs both of length a and density 1 Find the following b and c can be deduced from a with no further calculation a its moment of inertia about a leg b its polar moment of inertia about the right angle vertex c its moment of inertia about the hypotenuse 3C 2 Find the center of mass of the region inside one arch …
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