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Section 6-5 : More Volume Problems

  1. Use the method of finding volume from this section to determine the volume of a sphere of radius \(r\).
  2. Find the volume of the solid whose base is the region bounded by \(x = 2 - {y^2}\) and \(x = {y^2} - 2\) and whose cross-sections are squares with the base perpendicular to the \(y\)-axis. See figure below to see a sketch of the cross-sections.
    There is a standard xy-axis system shown in this sketch.  Also shown are the graphs of $x=2-y^{2}$ and $x=y^{2}-2$.  Both graphs start at their respective vertexes and then go left/right respectively until they hit the y-axis and then stop.  There is also a square that rises up out of the graph and is perpendicular to the positive y-axis.  The bottom right/left points of the square are on the curves.
  3. Find the volume of the solid whose base is a disk of radius \(r\) and whose cross-sections are rectangles whose height is half the length of the base and whose base is perpendicular to the \(x\)-axis. See figure below to see a sketch of the cross-sections (the positive \(x\)-axis and positive \(y\)-axis are shown in the sketch).
    There is a non-standard xy-axis system shown in this sketch.  The positive y-axis is horizontal and goes to the right out of the origin while the positive x-axis is vertical and goes down out of the origin.  A circle of radius r is also shown on the graph and it is noted that its equation is $x^{2}+y^{2}=r^{2}$.  There is also a rectangle that rises up out of the graph and is perpendicular to the positive x-axis.  The bottom right/left points of the rectangle are on the circle.
  4. Find the volume of the solid whose base is the region bounded by \(y = {x^2} - 1\) and \(y = 3\) and whose cross-sections are equilateral triangles with the base perpendicular to the \(y\)-axis. See figure below to see a sketch of the cross-sections.
    There is a standard xy-axis system shown in this sketch.  Also shown are the graphs of $t=x^{2}-1$ and $y=3$.  The graph of the parabola starts at its vertex and goes up until it hits the graph of $y=3$.  There is also a triangle that rises up out of the graph and is perpendicular to the positive y-axis.  The base right/left points of the triangle are on the graph of the parabola.
  5. Find the volume of the solid whose base is the region bounded by \(x = 2 - {y^2}\) and \(x = {y^2} - 2\) and whose cross-sections are the upper half of the circle centered on the \(y\)-axis. See figure below to see a sketch of the cross-sections.
    There is a standard xy-axis system shown in this sketch.  Also shown are the graphs of $x=2-y^{2}$ and $x=y^{2}-2$.  Both graphs start at their respective vertexes and then go left/right respectively until they hit the y-axis and then stop.  There is also a graph of the upper half of a circle that rises up out of the graph and is perpendicular to the positive y-axis.  The right/left points of the diameter of the semi-circle are on the curves.
  6. Find the volume of a wedge cut out of a “cylinder” whose base is the region bounded by \(y = \cos \left( x \right)\) and the \(x\)-axis between \( - \frac{\pi }{2} \le x \le \frac{\pi }{2}\). The angle between the top and bottom of the wedge is \(\frac{\pi }{4}\). See the figure below for a sketch of the “cylinder” and the wedge (the positive \(x\)-axis and positive \(y\)-axis are shown in the sketch).
    This image has two parts.  Both parts have a non-standard xy-axis system shown.  The positive y-axis is horizontal and goes to the right out of the origin while the positive x-axis is vertical and goes down out of the origin.  On the left is a 3D solid that looks almost like half of a cylinder.  At the bottom of the solid is the part of the graph of $y=\cos(x)$ in the domain $-\frac{\pi}{2} \le x \le \frac{\pi}{2}$.  The “hump” of the graph is the rightmost point on the y-axis.  The solid then rises out of this region and gives a shape that is like a half cylinder except its cross sections are not semi-circles but instead the region at the base.  Also shown in the left sketch is the top of the wedge that is inside the solid and it is shown that the top of the wedge makes an angle of $\frac{\pi}{4}$ with the base of the solid.  On the right side is just the wedge itself without the rest of the solid above it.  The “pointed” edge of the wedge is on the x-axis and the rounded side of the wedge again is in the shape of the graph of $\cos(x)$.  The top of the wedge makes an angle of $\frac{\pi}{4}$ with the base of the wedge.
  7. For a sphere of radius \(r\) find the volume of the cap which is defined by the angle \(\varphi \) where \(\varphi \) is the angle formed by the \(y\)-axis and the line from the origin to the bottom of the cap. See the figure below for an illustration of the angle \(\varphi \).
    This is an image of the full sphere.  It is “centered” at the origin of a typical xy-axis system.  The y-axis rises out of the top of the sphere and the x-axis moves out of the right side of the sphere.  The “cap” of the sphere is shown in a different color to illustrate it.  There is a line drawn out of the center of the sphere, which is also the origin of the xy-axis system, to where the base of the cap intersects the sphere.  The angle between the positive y-axis and this line is give as $\varphi $.