Find the area between \(y=x\) and \(y=x^2\) on the interval \(0 \le x \le 1\).

Find the area enclosed by \(y=4-x^2\) and the \(x\)-axis.

Find the area enclosed by \(y=x+2\) and \(y=x^2\).

Find the area enclosed by \(x=y^2\) and \(x=2-y\).

Find the area between \(y=\sin x\) and \(y=\cos x\) on \(0 \le x \le \pi/2\).

Find the area enclosed by \(y=x^2\) and \(y=2x-x^2\).

Find the total area between \(y=x^3\) and \(y=x\) on \(-1 \le x \le 1\).

Find the area between \(y=e^x\), \(y=1\), \(x=0\), and \(x=\ln 3\).

For a real constant \(a\) with \(a\ge 1\), the area between \(y=a\) and \(y=x^2\) on \(-1 \le x \le 1\) is \(4\). Find \(a\).

A learner finds the area between \(y=x\) and \(y=x^2\) on \(0 \le x \le 2\) by computing \(\int_0^2(x-x^2)\,dx=-2/3\). Explain the error and find the correct total area.