91Ó°ÊÓ

Solve the given problems by integration.A hot metal rod with an initial temperature of \(425^{\circ} \mathrm{C}\) is placed in a room with temperature \(20^{\circ} \mathrm{C}\). The time \(t\) (in min) required for the temperature \(T\) of the rod to cool to \(175^{\circ} \mathrm{C}\) is given by $$t=8.72 \int_{175}^{425} \frac{1}{T-20.0} d T$$.Find this time.

Solve the given problems by integration. Find the root-mean-square current in a circuit from \(t=0\) s to \(t=0.50 \mathrm{s}\) if \(i=i_{0} \sin t \sqrt{\cos t}\)

Solve the given problems by integration.In determining the temperature that is absolute zero ( 0 K, or about \(\left.-273^{\circ} \mathrm{C}\right),\) the equation \(\ln T=-\int \frac{d r}{r-1}\) is used. Here, \(T\) is the thermodynamic temperature and \(r\) is the ratio between certain specific vapor pressures. If \(T=273.16 \mathrm{K}\) for \(r=1.3361,\) find \(T\) as a function of \(r\) (if \(r>1\) for all \(T\) ).

Solve the given problems by integration.The time \(t\) and electric current \(i\) for a certain circuit with a voltage \(E,\) a resistance \(R,\) and an inductance \(L\) is given by \(t=L \int \frac{d i}{E-i R}\) If \(t=0\) for \(i=0,\) integrate and express \(i\) as a function of \(t\).

Solve the given problems by integration. In the study of the rate of radiation by an accelerated charge, the following integral must be evaluated: \(\int_{0}^{\pi} \sin ^{3} \theta d \theta\) of the integral.

Solve the given problems by integration. In finding the volume of a special O-ring for a space vehicle, the \(\int \frac{\sin ^{2} \theta}{\cos ^{2} \theta} d \theta\) must be evaluated. Perform this integration.

Solve the given problems by integration.Conditions are often such that a force proportional to the velocity tends to retard the motion of an object moving through a resisting medium. Under such conditions, the acceleration of a certain object moving down an inclined plane is given by \(20-v\). This leads to the equation \(t=\int \frac{d v}{20-v}\). If the object starts from rest, find the expression for the velocity as a function of time.

Solve the given problems by integration.An architect designs a wall panel that can be described as the firstquadrant area bounded by \(y=\frac{50}{x^{2}+20}\) and \(x=3.00 .\) If the area of the panel is \(6.61 \mathrm{m}^{2}\), find the \(x\) -coordinate (in \(\mathrm{m}\) ) of the centroid of the panel.

Solve the given problems by integration. For a voltage \(V=340 \sin 120 \pi t,\) show that the root-mean-square voltage for one period is \(240 \mathrm{V}\)

Solve the given problems by integration. For a current \(i=i_{0} \sin \omega t,\) show that the root-mean-square current for one period is \(i_{0} / \sqrt{2}\)