Interpreting The Definite Integral As Area

Question

A researcher computes the definite integral $$\int_{-2}^{0} (-x^2)\,dx$$ and obtains $$-\frac{8}{3}$$. The researcher concludes that the area of the region bounded by the graph of $$f(x) = -x^2$$ and the x-axis on the interval $$[-2, 0]$$ is negative. What is the error in this conclusion?

Accepted Answer

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 $$\frac{8}{3}$$.

Answer

The mistake is in choosing the incorrect limits of integration.

Answer

There is no mistake because the definite integral of a function that is negative over an interval is indeed negative.

Answer

The error is in the integration process, which should have produced a positive value directly.

Interpreting The Definite Integral As Area

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