MHB How to evaluate a double integral over a bounded region?

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To evaluate the double integral of 2x + y dA over the region R bounded by the lines y = 3x, 2x + y = 5, and the axes x = 0 and y = 0, it is crucial to accurately define the bounded region. The discussion highlights confusion regarding the boundaries, noting that the region forms a triangle in the first quadrant where both x and y are positive. There is skepticism about the correctness of the problem as presented, particularly regarding the provided answer of -2875/6, which seems inconsistent with the integral's expected positive value. The participants agree that the teacher may have made an error in framing the question. The evaluation of the double integral requires careful consideration of the region's geometry and the nature of the function being integrated.
mduffy
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how do I evaluate the double integral 2x + y dA over the region R bounded by y = 3x, 2X + y = 5, x = 0, and y = 0
 
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mduffy said:
how do I evaluate the double integral 2x + y dA over the region R bounded by y = 3x, 2X = y = 5, x = 0, and y = 0

Check this entry ...
 
skeeter said:
Check this entry ...
Thanks...should be 2x + y = 5
 
over the region R bounded by y = 3x, 2X + y = 5, x = 0, and y = 0

Region R is bounded by either x = 0 or y = 0 ... how is it bounded by both axes and the two given lines?
 
skeeter said:
Region R is bounded by either x = 0 or y = 0 ... how is it bounded by both axes and the two given lines?

Agreed! I think the teacher made a mistake with the problem. The given answer (no steps shown) is -2875/6. I am thinking there is an error in how the question is asked.
 
The lines y= 3x and 2x+ y= 5 form one triangle with x= 0 and another with y= 0. But both of those are in the first quadrant where x and y have only positive values. The integral of (2x+ y) dA can't be negative over either of them.
 

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