finding area using integration Watch

Iconic_panda
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so I know that if we integrate between two x-coordiates you will get an area. if that area is below the x-axis, the area will appear as a negative value. Would that happen to parametric and polar integration as well? If i integrate an area below x-axis using parametric and polar coordinates, would it appear as positive value?
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RDKGames
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(Original post by Iconic_panda)
so I know that if we integrate between two x-coordiates you will get an area. if that area is below the x-axis, the area will appear as a negative value. Would that happen to parametric and polar integration as well? If i integrate an area below x-axis using parametric and polar coordinates, would it appear as positive value?
All of those can be answered in one go if you consider what an integral actually is for an area. For \displaystyle \int_a^b f(x) .dx you are essentially summing up the products f(x) \cdot dx from x=a to x=b. If a<b we get that dx>0. The only way the product can be -ve is if f(x) < 0 in the domain, hence how we get -ve area.

Same principle applies to any other integral. For polar, we have \displaystyle \int_{\theta_1}^{\theta_2} r(\theta) . d\theta and if  r(\theta) \geq 0 then it cannot be negative. Generally, we have \theta_1 < \theta_2 so dx is +ve, and the product is +ve and so the area is always +ve. You can get it being -ve if the larger limit is the lower limit so that dx is -ve, or is r(\theta)<0
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