double_integrals

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3. Double Integrals 3A. Double Integrals in Rectangular Coordinates 3A-1 Evaluate each of the following iterated integrals: (6x2 + 2y) dy dx b) lnI2 Ln(, 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 - /L x2 and the x-axis. dy dx ii) 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. dy dx ii) dx dy d)* R is the finite region lying between the parabola y2 = x and the line through (2,O) having slope 1. dy dx ii) 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) x dA; R is the finite region bounded by the axes and 2y + x = 2 b) /L(2x + Y2) dA; R is the finite region in the first quadrant bounded by the axes and = 1 - x; (dx dy is easier). c) y dA; R is the triangle with vertices at (f 1, 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
2 E. 18.02 EXERCISES 3A-5

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## This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.

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double_integrals - MIT OpenCourseWare http:/ocw.mit.edu...

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