Question

Let i be the region enclosed by the planes x = 0,
paraboloid z = x2 + y2. Compute the integral
SIT and
4xdV
by sketching the region of integration. Explain yo​

Answer 1

Answer:

Step-by-step explanation:

When we defined the double integral for a continuous function in rectangular coordinates—say,  g  over a region  R  in the  xy -plane—we divided  R  into subrectangles with sides parallel to the coordinate axes. These sides have either constant  x -values and/or constant  y -values. In polar coordinates, the shape we work with is a polar rectangle, whose sides have constant  r -values and/or constant  θ -values. This means we can describe a polar rectangle as in Figure 5.28(a), with  R={(r,θ)|a≤r≤b,α≤θ≤β}.  

In this section, we are looking to integrate over polar rectangles. Consider a function  f(r,θ)  over a polar rectangle  R.  We divide the interval  [a,b]  into  m  subintervals  [ri−1,ri]  of length  Δr=(b−a)/m  and divide the interval  [α,β]  into  n  subintervals  [θi−1,θi]  of width  Δθ=(β−α)/n.  This means that the circles  r=ri  and rays  θ=θi  for  1≤i≤m  and  1≤j≤n  divide the polar rectangle  R  into smaller polar subrectangles  Rij  (Figure 5.28(b)).