APFIVE A researcher calculates the area under the curve of $$f(x) = -x^2$$ over the interval $$[-2, 0]$$ by computing the definite integral $$\int_{-2}^{0} (-x^2)\,dx$$ and obtains $$-\frac{8}{3}$$, then concludes that the area is negative. What is the error in this process?
The mistake is in choosing the incorrect limits of integration.
The error is in interpreting the definite integral as the area under the curve; when the function is below the x-axis, the area should be taken as the absolute value, namely $$\frac{8}{3}$$.
There is no mistake because the definite integral of a function that is negative over an interval is indeed negative.
The error is in the integration process, which should have produced a positive value directly.
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