ARE YOU FINDING THE RIGHT AREA?!! Area under vs. area enclosed by a graph

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Often calculus textbooks and professors will distinguish between area under the curve, or area under the graph, and the area enclosed by the curve.

The area under the curve is the net area, it's the value of the integral. This kind of area treats all of the area above the x-axis as positive, and all of the area below the x-axis as negative. Therefore, in this case, if there is more area above the x-axis than below it, then net area will be positive. But if there is more area below the x-axis than above it, then net area will be negative.

Contrast this with the area enclosed by the curve, where you're supposed to treat all area as positive, or take the absolute value of the area. In this case, it wouldn't matter how much area was below or above the x-axis, the area enclosed by the curve would always be positive.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”