advertisement

Homework week 10 Problems. 1. Consider region R given by the Semi-circle of radius r 1 in the upper-half of the xy-plane. Calculate the integral of the function 2 2 f xy ex y over that given region R. Hint: change to polar coordinates. remember that: Solution: The integral changes like e x r cos θ y r sin θ dydx rdrdθ x2 y2 2 er rdrdθ dydx R R the limits of integration come easily from the sketch of the region R, Drawing a radial line we notice y y= 1−x2 r=1 1 θ=π θ=0 −1 Figure 0.1: 0 that the r limits of integration go from r to θ π so the integral changes to e x2 r 0 to r y2 x 1 0 1, 0 θ 1, while the θ-limits of integration go from θ dydx R π π 0 1 2 er rdrdθ 0 π 1 r2 e 2 1 dθ 1 e 1 dθ 2 π e 1 2 0 π 0 1 0 0 Notice to go from row 1 to row 2 we used the change of variable so 1 r2 e rdr r2 v dv 2 rdr ev dv ev er 2 0 e 1 1 0 2. Find the volume of the following figure: Solution: z 1 y=x 2 1 1 y x Figure 0.2: fig1 Drawing a vertical line we find that the z-limits are: 01 dz. Drawing a line parallel to the y-axis we find the y-limits: x12 dy and finally sliding the last line in the direction: ∞ to ∞ of the x-axis, we find that: 1 0 . Therefore the volume of this figure is given by z 1 V 1 0 x2 0 1 1 0 x2 1 1 y x2 0 1 0 1 1 dzdydx 1 0 dydx dydx 1 x2 dx 1 x dx 2 13 x 1 2 1 3 3 0 1 3 x 0 3. Read problems 8.4 a) in these problems the volume was found by using double integrals methods. Using triple integrals find the volumes of problems 8.4 a). Solution: 2 In this case drawing a vertical line will indicate that the z-limit goes from: 05 x y dz while the y and x limits will remain the same as in 8.4 a). So the answer to this problem is 1 V 0 5 x2 y x 1 0 0 2 dzdydx the solution of the integral is as in problem 8.4 a) once you perform the first integral. 3