91Ó°ÊÓ

Evaluate the indefinite integral. $$ \int \frac{d x}{x^{4} \sqrt{16+x^{2}}} $$

Use the prismoidal formula to find the exact volume of the given solid. A right-circular cylinder of height \(h\) and base radius \(r\).

Evaluate the definite integral. $$ \int_{3}^{4} \frac{\left(5 x^{3}-4 x\right) d x}{x^{4}-16} $$

The integral \(\int_{0}^{2} e^{-x^{2}} d x\) is very important in mathematical statistics. It is called a "probability integral" and it cannot be evaluated exactly in terms of elementary functions. Use the trapezoidal rule with \(n=6\) to find an approximate value and express the result to three decimal places.

Show that the exact value of \(\int_{0}^{2} \sqrt{4-x^{2}} d x\) is \(\pi\). Approximate the definite integral by the trapezoidal rule to three decimal places, with \(n=8\), and compare the value so obtained with the exact value.

Find the centroid of the solid of revolution obtained by revolving about the \(x\) axis the region bounded by the curve \(y=\sin x\), the \(x\) axis, and the line \(x=\frac{1}{2} \pi\).

At any instant the rate at which a substance dissolves is proportional to the product of the amount of the substance present at that instant and the difference between the concentration of the substance in solution at that instant and the concentration of the substance in a saturated solution. A quantity of insoluble material is mixed with \(10 \mathrm{lb}\) of salt initially, and the salt is dissolving in a tank containing 20 gal of water. If \(5 \mathrm{lb}\) of salt dissolves in \(10 \mathrm{~min}\) and the concentration of salt in a saturated solution is \(3 \mathrm{lb} / \mathrm{gal}\), how much salt will dissolve in \(20 \mathrm{~min} ?\)