Consider the function f(x, y) = 2+cos(y) which is well-defined and smooth for all (x, y)...

 

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Consider the function f(x, y) = 2+cos(y) which is well-defined and smooth for all (x, y) = R². Now imagine the simple domain D which is bounded by the unit interval (0, 1) on the x-axis from below and by the graph of two times the arcus-tangens from above, which means exactly: = {(x, y) = R²|x€ (0,1) ^ 0 <y < 2 arctan(x)}. Now try to compute the integral ff(x, y) dxdy ! i.e. the indefinite integral f 2+cos(p) dy you have to fight here a 2+cos(y) Hint: Follow your spontaneous first idea, to apply Cavalieri's principle - simply slicing the domain D with respect to the x-axis from 0 to 1. In order to find the "primitive" of the function y little bit. Introduce the new variable t =tan(y/2) and then use the fact that the derivative of ytan(y) is the function y cos(u) together with the well-known trigonometric formula for "cos(a + 3)", in order to discover the equations: and dt = (1+1²) dy and These equations certainly help you to crack the "inner integral of f w.r.t. y", and you may just use them if you cannot prove them. 1+²= 1 cos² (y/2) 1-1² 1+12 = cos(y). Consider the function f(x, y) = 2+cos(y) which is well-defined and smooth for all (x, y) = R². Now imagine the simple domain D which is bounded by the unit interval (0, 1) on the x-axis from below and by the graph of two times the arcus-tangens from above, which means exactly: = {(x, y) = R²|x€ (0,1) ^ 0 <y < 2 arctan(x)}. Now try to compute the integral ff(x, y) dxdy ! i.e. the indefinite integral f 2+cos(p) dy you have to fight here a 2+cos(y) Hint: Follow your spontaneous first idea, to apply Cavalieri's principle - simply slicing the domain D with respect to the x-axis from 0 to 1. In order to find the "primitive" of the function y little bit. Introduce the new variable t =tan(y/2) and then use the fact that the derivative of ytan(y) is the function y cos(u) together with the well-known trigonometric formula for "cos(a + 3)", in order to discover the equations: and dt = (1+1²) dy and These equations certainly help you to crack the "inner integral of f w.r.t. y", and you may just use them if you cannot prove them. 1+²= 1 cos² (y/2) 1-1² 1+12 = cos(y).

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First well set up the integral using Cavalieris principle by slicing the domain ... View the full answer