91Ó°ÊÓ

Solve the given problems by integration. The vertical cross section of a highway culvert is defined by the region within the ellipse \(1.00 x^{2}+9.00 y^{2}=9.00,\) where dimensions are in meters. Find the area of the cross section of the culvert.

Solve the given problems by integration. Using the identity \(\cos \alpha \cos \beta=\frac{1}{2}[\cos (\alpha+\beta)+\cos (\alpha-\beta)]\) integrate \(\int \cos 3 x \cos 4 x d x\)

Solve the given problems by integration. Under specified conditions, the time \(t\) (in min) required to form \(x\) grams of a substance during a chemical reaction is given by \(t=\int d x /[(4-x)(2-x)] .\) Find the equation relating \(t\) and \(x\) if \(x=0\) g when \(t=0\) min.

Solve the given problems by integration. The general expression for the slope of a curve is \(d y / d x=x^{3} \sqrt{1+x^{2}} .\) Find an equation of the curve if it passes through the origin.

Solve the given problems by integration.Show that \(\int_{0}^{3} \frac{d x}{2 x+2}=\ln 2\).

Solve the given problems by integration. If the current \(i\) (in \(\mathrm{A}\) ) in a certain electric circuit is given by \(i=110 \cos 377 t,\) find the expression for the voltage across a \(500-\mu F\) capacitor as a function of time. The initial voltage is zero. Show that the voltage across the capacitor is \(90^{\circ}\) out of phase with the current.

Solve the given problems by integration. Integrate \(\int \sqrt{\frac{1+x}{1-x}} d x\) by first multiplying the numerator and denominator of the fraction under the radical by \(1+x\).

Solve the given problems by integration.The general expression for the slope of a curve is \(\frac{\sin x}{3+\cos x}\). If the curve passes through the point \((\pi / 3,2),\) find its equation.

Using the identity \(\cos \alpha \cos \beta=\frac{1}{2}[\cos (\alpha+\beta)+\cos (\alpha-\beta)]\) integrate \(\int \cos 3 x \cos 4 x d x\) Find the volume generated by revolving the region bounded by \(y=\sin x\) and \(y=0,\) from \(x=0\) to \(x=\pi,\) about the \(x\) -axis.

Solve the given problems by integration. Find the area bounded by \(y=x \sin x,\) the \(x\) -axis, (a) between 0 and \(\pi,\) (b) between \(\pi\) and \(2 \pi,(\mathrm{c})\) between \(2 \pi\) and \(3 \pi .\) Note the pattern.