Question

Avni asserts that \(\int_{-2}^{1} 2 \mathrm{x}^{2} \mathrm{dx}\) is obviously positive. "After all, the integrand is never negative and \(-2<1 . "\) "You are wrong again," Meet replies, "It's negative. Here are my computations. Let \(\mathrm{u}=\mathrm{x}^{2}\); hence \(\mathrm{du}=2 \mathrm{x} \mathrm{dx}\). Then \(\int_{-2}^{1} 2 x^{2} d x=\int_{-2}^{1} x \cdot 2 x d x\) \(=\int_{4}^{1} \sqrt{\mathrm{u}} \mathrm{du}=-\int_{4}^{1} \sqrt{\mathrm{u}} \mathrm{du}\)

Short answer

Answer: The definite integral of \(2x^2\) over the interval \([-2, 1]\) using Meet's substitution method is \(-\frac{14}{3}\).

Step-by-step solution

Consider Meet's substitution method

Meet let \(u = x^2\) and then \(du = 2x dx\). By replacing \(2x dx\) with \(du\) and \(x^2\) with \(u\), Meet's substituting inner expression becomes \(\int_{-2}^{1} x \cdot 2x dx = \int_{4}^{1} \sqrt{u} du\). Now let's analyze the limits of integration.

Analyze the limits of integration

Meet changed the limits of integration from \([-2,1]\) to \([4,1]\). This is incorrect. When applying substitution, one must change the limits according to the substitution variable. In this case, when \(x=-2\), \(u=(-2)^2=4\). When \(x=1\), \(u=(1)^2=1\). Therefore, the correct limits of integration should be \([4,1]\) to match the variable change.

Apply the correct limits of integration

Now let's re-write Meet's integral with the correct limits of integration: \(\int_{4}^{1} \sqrt{u} du\). In order to make the limits of integration go from lower to higher, we can change the order of limits and add a negative sign in front: \(-\int_{1}^{4} \sqrt{u} du\).

Integrate with respect to u

Now we can evaluate the definite integral. To do so, we need to find the antiderivative of \(\sqrt{u}\) with respect to \(u\) and then evaluate it at the limits \(1\) and \(4\). The antiderivative of \(\sqrt{u}\) is \(\frac{2}{3}u^{\frac{3}{2}}\). So we have: $$-\int_{1}^{4} \sqrt{u} du = -\left(\frac{2}{3}u^{\frac{3}{2}}\right)\Big|_{1}^{4}$$

Evaluate the definite integral

Now we evaluate the antiderivative at \(1\) and \(4\) and subtract the results: $$-\left(\frac{2}{3}u^{\frac{3}{2}}\right)\Big|_{1}^{4} = -\left[\left(\frac{2}{3}(4)^{\frac{3}{2}}\right) - \left(\frac{2}{3}(1)^{\frac{3}{2}}\right)\right]$$ $$= -\left[\frac{2}{3}(8) - \frac{2}{3}\right] = -\frac{14}{3}$$ Now we can see that, when applying Meet's substitution method with the correct limits of integration, the definite integral does indeed evaluate to a negative number: \(-\frac{14}{3}\). This is contrary to Avni's assertion. It's essential to always check our work carefully, especially when dealing with definite integrals and substitution, to ensure we get accurate results.