Question

Evaluate the double integral ∬Dx^2dA,
where D={(x,y):1≤x≤e,0≤y≤lnx

Answer 1

To evaluate the double integral ∬Dx^2dA over the region D={(x,y):1≤x≤e,0≤y≤lnx}, we first need to set up the limits of integration.

The region D is defined as the set of points (x, y) where 1≤x≤e and 0≤y≤lnx. In terms of a rectangular coordinate system, this means that D is a rectangle in the first quadrant, bounded by the x-axis, the line y=lnx, and the vertical lines x=1 and x=e.

We can express the limits of integration as follows:

∬Dx^2dA = ∫[a,b]∫[c,d]x^2dydx

where the limits of integration [a,b] and [c,d] are determined by the boundaries of the region D.

In this case, the boundaries of D are:
- The x-axis (y=0) as the lower boundary for y
- The line y=lnx as the upper boundary for y
- The vertical line x=1 as the lower boundary for x
- The vertical line x=e as the upper boundary for x

Therefore, the limits of integration are:
a = 1
b = e
c = 0
d = lnx

Putting it all together, we can express the double integral as:

∬Dx^2dA = ∫[1,e]∫[0,lnx]x^2dydx

Now, we can proceed to evaluate the double integral.