91Ó°ÊÓ

Calculate by double integrals the area bounded by each of the following pairs of curves: (a) \(y^{2}=x^{3}\) and \(y=x\). (b) \(x^{2}+y^{2}=10\) and \(y^{2}=9 x\). (c) \(y^{2}=x+1\) and \(x+y=1\). (d) \(y=9-x^{2}\) and \(y=x+7\). (e) \(x y=4\) and \(x+y=5\). (f) \(y^{2}=5-x\) and \(y^{2}=4 x\). (g) \(y=2 x-x^{2}\) and \(y=3 x^{2}-6 x\). (h) \(4 y=x^{3}\) and \(y=x^{3}-3 x\).

Find the area inside the circle \(\rho=2 a \cos \theta\) and outside the circle \(\rho=\) a. Ans. \(a^{2}\left(\frac{\pi}{3}+\frac{\sqrt{3}}{2}\right)\).

(a) Calculate the moment of inertia of a uniform, thin, circular disk of radius \(a\) and total mass \(M\) when the axis of rotation is perpendicular to the plane of the disk and through its center. Ans. \(\frac{1}{2} M a^{2}\). (b) Calculate the moment of inertia of a uniform solid circular cylinder of total mass \(M\), radius \(a\), and length \(l\) about its axis. Suggestion: The cylinder can be regarded as the sum of a large number of thin disks.

Find the volume bounded by the elliptic paraboloid \(z=1-\left(x^{2} / a^{2}\right)-\) \(\left(y^{2} / b^{2}\right)\) and the \(x y\)-plane. Ans. \(\pi a b / 2\).

Express as a triple integral and evaluate the following: (a) The volume in the first octant bounded by the coordinate planes and the plane \((x / a)+(y / b)+(z / c)=1\). Ans. \(a b c / 6\). (b) The volume bounded by the paraboloid \(y^{2}+z^{2}=x+1\) and the plane \(x=0\). (c) The volume in the first octant bounded by the surfaces \(x^{2}+z=1\), \(y^{2}+z=1, x=0, y=0\), and \(z=0 .\) Ans. \(\frac{1}{2}\) (d) The volume of one of the wedges cut out from the cylinder \(x^{2}+y^{2}\) \(=r^{2}\) by the planes \(z=0\) and \(z=m x\). (e) The volume bounded by the cylinder \(x^{2}+y^{2}=4\) and the planes \(y+\) \(z=4\) and \(z=0\). Ans. \(16 \pi\). (f) The volume bounded by the paraboloid \(y^{2}+z^{2}=4 a x\), the parabolic cylinder \(y^{2}=a x\), and the plane \(x=3 a\). (g) The volume over the area common to the two parabolas \(x=y^{2}\) and \(y=x^{2}\) and under the surface \(z=12+y-x^{2}\). Ans. \(\frac{569}{140}\) (h) The volume bounded by the two paraboloids \(z=8-x^{2}-y^{2}\) and \(z=\) \(x^{2}+3 y^{2}\) (i) The volume bounded by the paraboloid \(z=2 x^{2}+y^{2}\) and the cylinder \(z=4-y^{2} . \quad\) Ans. \(4 \pi\).

Find the mass of a lamina bounded by \(x=0, x=1, y=0\), and \(y=e^{x}\) if the density at any point varies as the distance of that point from the \(x\) axis.

(a) Show that the sum of the moments of inertia of any uniform plane laminar body (thin disk of arbitrary bounding shape) about any two perpendicular axes in the plane of the body is equal to the moment of inertia about an axis through the point of intersection of the two axes and perpendicular to the plane of the lamina. (b) Apply the result of part (a) to find the moment of inertia of a uniform circular disk about a diameter.

Find the volume under the surface \(z=x y\) and above the area in the first quadrant bounded by \(y=0, y=x\), and \(x^{2}+y^{2}=1\).

Find the volume under the plane \(z=x+y\) and above the area in the first quadrant bounded by the ellipse \(4 x^{2}+9 y^{2}=36\).

Find the volume in the first octant under the ellipsoidal surface \(9 \rho^{2}+\) \(4 z^{2}=36\).