Evaluate the double integral by first identifying it as the volume of a solid: If R is the rectangle R={(x,y)|0<=x<=5,0<=y<=3}, then _(R)5-xdA=,(input a most symplified fraction) The double integral equals the volume of the solid that lies below the plane z=(input a function, place the constant term before any variable term) and above the rectangle [0,6]\times [0,3], as shown in the figure. (Hint: A more regular solid with twice that volume should be easy to see from the figure.) The integrand is not a constant, so you cannot bring the integrant out of the integral. To find the integral without resorting to volumes, you can express the integral as an iterated integral, and then integrate with respect to x and y, consecutively. The integration order does not matter here since the domain is a rectangular region: _(R)5-xdA=\int_0^a \int_0^b 5-xdxdy=\int_0^c \int_0^d 5-xdydx, where a=,b= type your answer... d=1 After integration of either one, you get the integral value which should be the same as what you got above via volume. If you did not see how to use volume, this more elementary integration approach (it is just doing 1-d integration twice) will also provide you with the correct answer. A third way to get the integral is via "summation". Normally, Riemann sum only gives you an approximate value of the integral, but when the integrand is linear, and if you choose the grid points wisely by placing them in the middle, you can get the exact integral. Here, since the integrand is linear (a plane), you can indeed try Riemann sum to see if it can give you an exact integral. Since the integrand is linear on the whole region, you do not even need to do a sum, you can choose to use just one grid point, by placing it in the middle of the whole region, which is (x_(0),y_(0))=((5)/(2),(3)/(2)), then the one-point Riemann sum leads to _(R)f(x,y)dA~~f(x_(0),y_(0))_(R)dA Note that here f(x,y)=5-x, so you get f(x_(0),y_(0))=(input a most simplified fraction). And _(R)dA= Does the "approximate" one-point Riemann sum actually give you the exact integral?