Sketching Region of Integration of unspecified function?

[PsychonautQQ]

Homework Statement

Double integral (The first one is lower bound 0 and upper bound 1, the second one is lower bound 7x and top one 7). of f(x,y)dydx and my teacher wants me to sketch the region of integration. Then reverse the area of integration.

The Attempt at a Solution

So I was thinking about this and trying to figure out how to do this with an unspecified function.. Could I just show in random functions like (xy) or (x^2)(y^2) and look for some sort of pattern? Am I on the right track here as far as the methodology of solving the problem goes?

 

[Mark44]

Homework Statement

Double integral (The first one is lower bound 0 and upper bound 1, the second one is lower bound 7x and top one 7). of f(x,y)dydx and my teacher wants me to sketch the region of integration. Then reverse the area of integration.

The Attempt at a Solution

So I was thinking about this and trying to figure out how to do this with an unspecified function.. Could I just show in random functions like (xy) or (x^2)(y^2) and look for some sort of pattern? Am I on the right track here as far as the methodology of solving the problem goes?

No, you're not on the right track. The function in the integrand doesn't matter.

Here's your integral, laid out in LaTeX:
$$\int_{x = 0}^1 \int_{y = 7x}^7 f(x, y) dy~dx $$

The inner integral is with x held fixed and y varying; the outer integral is with x varying.

 

[LCKurtz]

And the way to do a problem like this is to sketch the region, as your teacher requested. Once you have the region sketched, set the integral up as a dxdy integral using the sketch. So what does your sketch look like and what did you get when you reversed the limits?

 

[HallsofIvy]

I am puzzled as to why you think the "unspecified function" is at all relevant when you are only asked to sketch the region and then reverse the limits. The only thing relevant are the limits of integration.
\int_{x=0}^1\int_{y= 7x}^7 f(x,y)dydx
tells you that x lies between 0 and 1 and, for each x y lies between 7x and 7.

So- on an xy- coordinates system, draw the vertical lies x= 0 (left boundary) and x= 1 (right boundary). Draw the horizontal line y= 7 (top boundary) and the line y= 7x (bottom boundary). The region of integration is inside those lines.

As for reversing the order, What is the smallest value of y inside that region? What is the largest? For each y what are the lower and upper bounds for x?