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The image shows a one-sheeted hyperboloid symmetric around the axis. The blue curve is the unique hyperboloid geodesic passing through the given point (shown in black) and intersecting the parallel (i.e. the circle of latitude) through that point at the given angle . t2 1sin ct 3 5 The hyperboloid of one sheet is a doubly ruled surface. Through each its points there are two lines that lie on the surface. Both kinds of circular hyperboloids as well as the cone can be included in one family of surfaces by modifying their de ning equations slightly. Consider the equations x2 + y 2= z + e where eis a constant. A quadric surface given by an equation of the form (x 2 / a 2 ) ± (y 2 / b 2 ) - (z 2 / c 2 ) = 1; in certain cases it is a hyperboloid of revolution, which can be realized by rotating the pieces of a hyperbola about an appropriate axis. For a vector normal to the plane we can choose the cross product of two vectors in the plane:4( 1) 3( 2) 4( 3) 0 4 3 4 is normal to the plane 4 3 4 10n i j kx y z8 6 8i j kx y z . and 2, 4, 1v= PR . Two distinct planes in 3-space either are parallel or intersect in a line. 7. (22 points) Sketch (or describe) the solid whose volume is given by a. 2 242 0 0 0 yy dxdzdy ³ ³ ³ b. 4 2 4 00r rdzd dr S ³³³ T c. 2 /4 3 2 0 0 0 sin SS ³ ³ ³U I U I Td d d 8. (12 points)Find the volume of the solid inside the sphere x y z2 2 2 9 and inside the cylinder xy22 1. Vrhdy if slices are horizontal Physical Applications: Physics Formulas Associated Calculus Problems Mass: Mass = Density * Volume (for 3‐D objects) Mass = Density * Area (for 2‐D objects) Mass = Density * Length (for 1‐D objects) Mass of a one‐dimensional object with variable linear density: () ().