Evaluate the double integral 4x^3 dA, where R is the region bounded by the graphs of y=(x-1)^2 and y=-x+3.

Question

idealist10 · Answer

@ganeshie8 @phi @wio @radar

idealist10 · Answer

Just tell me how to find the limits of integration for y. I can do rest of the problem. Because (x-1)^2=-x+3 so x=2, -1 for dx. But what about y?

phi · Answer

you could graph the two curves: a parabola and a line, using for example Geogebra

idealist10 · Answer

But what are the limits of integration for y?

phi · Answer

I would make the inner integral over y
the upper limit would be the blue line, and the lower limit the parabola
these would be functions of x

then the outer integral would be over x, with numeric limits

idealist10 · Answer

So the limits of integration for y are 1 to 4? But the answer to this problem is 72/5.

phi · Answer

no, the limits for y are functions of x .  in other words, depending on what x we are at, the y limits change (which we can see by looking at the graph)
$$ \int_{-1}^2 \int_{(x-1)^2}^{3-x} 4 x^3 dy\ dx $$

idealist10 · Answer

Thank you! Now I understand.