Section 3.1 Calculus 2 - Professor Clontz

3.1 Area Between Curves

Recall that \(\int_a^b f(x)\,dx\) is the net area between \(y=f(x)\) and \(y=0\).
Let \(f(x)\leq g(x)\) for \(a\leq x\leq b\). We define the area between the curves \(y=f(x)\) and \(y=g(x)\) from \(a\) to \(b\) to be the integral \(\int_a^b [g(x)-f(x)]\,dx\).
We call \(y=f(x)\) the bottom curve and \(y=g(x)\) the top curve.
Example Find the area between the curves \(y=2+x\) and \(y=1-\frac{1}{2}x\) from \(2\) to \(4\).

Example Find the area bounded by the curves \(y=x^2-4\) and \(y=8-2x^2\).
Example Prove that the area of a circle of radius \(r\) is \(\pi r^2\). (Hint: use the curves \(y=\pm\sqrt{r^2-x^2}\).)

Areas between functions \(f(y)\leq g(y)\) may be found similarly, but in this case \(x=f(y)\) is the left curve and \(x=g(y)\) is the right curve.
Example Find the area bounded by the curves \(y=\sqrt{x}\), \(y=0\), and \(y=x-2\).

Find the area between the curves \(y=4\) and \(y=4x^3\) from \(-1\) to \(1\).
Find the area bounded by the curves \(y=x^2-2x\) and \(y=x\).
Find the area bounded by the curves \(y=\pm\sqrt{4-x}\) and \(x=3\).
Find the area bounded by the curves \(y=0\), \(x=0\), \(y=1\), and \(y=\ln x\).
Use a definite integral to prove that the area of the triangle with vertices \((0,0)\), \((b,0)\), \((0,h)\) is \(\frac{1}{2}bh\).
Find the area of the ellipse \(9x^2+16y^2=25\).