In evaluating a double integral over a region D, a sum of iterated integrals was obtained as follows: ∬Df(x, y)dA=∫5_0∫(2/5)y_0 f(x, y)dxdy+∫7_5∫(7−y)_0 f(x, y)dxdy.

Sketch the region D and express the double integral as an iterated integral with reversed order of integration.

∫b_a∫g2(x)_g1(x) f(x, y)dydx

a= b=

g1(x)= g2(x)=

Answer

,

,  

Step-by-step explanation:

Given that

For the term  .

Limits for is from to and for is from to  and the region , for this double integration is the shaded region as shown in graph 1.

Now, reverse the order of integration, first integrate with respect to then with respect to . So, the limits of become from to and limits of become from to as shown in graph 2.

So, on reversing the order of integration, this double integration can be written as

Similarly, for the other term  .

Limits for is from to and limits for is from to  and the region , for this double integration is the shaded region as shown in graph 3.

Now, reverse the order of integration, first integrate with respect to then with respect to . So, the limits of become from to and limits of become from to as shown in graph 4.

So, on reversing the order of integration, this double integration can be written as

Hence, from equations , and , on reversing the order of integration, the required expression is

Now, compare the RHS of the equation with

We have,

and .