Worked Examples: Applications of Integration

Find the area between $y=x$ and $y=x^2$ on $[-1,2]$.

Find the area enclosed by $r=2\cos\theta$ for $-\pi/2\le\theta\le\pi/2$.

Rotate the region under $y=\sqrt{x}$ from $x=0$ to $x=4$ about the $x$-axis. Find its volume.

Rotate the region between $y=x$ and $y=x^2$ on $0\le x\le1$ about the $x$-axis.

Rotate the region under $y=2x-x^2$ on $0\le x\le2$ about the $y$-axis using cylindrical shells.

Find the arc length of $y=\frac23x^{3/2}$ from $x=0$ to $x=1$.

Find the surface area generated by revolving $y=x$ on $0\le x\le1$ about the $x$-axis.

Revolve $y=x^2$ for $0\le x\le1$ about the $y$-axis and find the surface area.

Find the centroid of the region between $y=\sqrt{x}$ and $y=x^2$ on $0\le x\le1$.

A right triangular area has base $b$ on the $x$-axis, height $h$ on the $y$-axis, and hypotenuse $y=h(1-x/b)$. Find $I_x$ and $I_y$ about the coordinate axes.

A uniform thin plate has area $A$ and mass $m$ with constant areal density $\sigma=m/A$. Relate its second moment of area about an axis to its mass moment of inertia about the same in-plane axis.

A conical tank, point down, has top radius 4 m and height 10 m. Water fills it to depth 8 m. Find the work to pump all water to the top rim. Use $\rho=1000\,\mathrm{kg/m^3}$ and $g=9.8\,\mathrm{m/s^2}$.

A vertical rectangular dam is 20 m wide and submerged from the free surface to depth 15 m. Find the resultant hydrostatic force and its depth of application. Use $\rho=1000\,\mathrm{kg/m^3}$ and $g=9.8\,\mathrm{m/s^2}$.

A circle of radius 2 is centered at $(3,0)$ and revolved about the $y$-axis. Use Pappus' theorem to find the volume and verify the theorem's geometry condition.

Work two independent applications. (1) Demand is $D(q)=50-q$, supply is $S(q)=10+q$, and equilibrium occurs where they are equal. Find consumer and producer surplus. (2) Let $f(x)=2x$ on $0\le x\le1$ and zero elsewhere. Verify it is a PDF and find $P(X>0.6)$.