91Ó°ÊÓ

Find the area of the surface generated by revolving the curve \(x=t+\sqrt{7}, y=t^{2} / 2+\sqrt{7} t\), for \(-\sqrt{7} \leq t \leq \sqrt{7}\) about the \(y\) -axis.

The slope of the tangent line to the hyperbola $$ 2 x^{2}-7 y^{2}-35=0 $$ at two points on the hyperbola is \(-\frac{2}{3}\). What are the coordinates of the points of tangency?

Find the area of the surface generated by revolving the curve \(x=t^{2} / 2+a t, y=t+a\), for \(-\sqrt{a} \leq t \leq \sqrt{a}\) about the \(x\) -axis.

The ends of an elastic string with a knot at \(K(x, y)\) are attached to a fixed point \(A(a, b)\) and a point \(P\) on the rim of a wheel of radius \(r\) centered at \((0,0) .\) As the wheel turns, \(K\) traces a curve \(C\). Find the equation for \(C\). Assume that the string stays taut and stretches uniformly (i.e., \(\alpha=|K P| /|A P|\) is constant).

Find the equations of the tangent lines to the ellipse \(x^{2}+2 y^{2}-2=0\) that are parallel to the line $$ 3 x-3 \sqrt{2} y-7=0 $$

Sketch the reciprocal spiral given by \(r=c / \theta .\) For \(c>0\), does it unwind in the clockwise direction?

Evaluate the integrals . $$ \int_{0}^{1}\left(x^{2}-4 y\right) d x, \text { where } x=t+1, y=t^{3}+4 $$

Find the area of the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\).

Name the conic \(y^{2}=L x+K x^{2}\) according to the value of \(K\) and then show that in every case \(|L|\) is the length of the latus rectum of the conic. Assume that \(L \neq 0\).

Find the volume of the solid obtained by revolving the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\) about the \(y\) -axis.