2.2E: Exercises

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Reading Questions

What is the definition of a solid's cross-section?
What is the general definite integral formula for the volume of a solid using the slicing method, where $A(x)$ is the cross-sectional area at $x$?
List the three steps in the problem-solving strategy for finding volumes by the slicing method.
In Example 1.2.1, what geometric principle is used to find the side length $s$ of the square cross-section in terms of its distance $x$ from the apex of the pyramid?
If the cross-sections of a solid are taken perpendicular to the y-axis, what variable must the area function and the integration be in terms of?
According to the text, what is the mathematical definition of a cylinder? Does its base have to be a circle?

1) Derive the formula for the volume of a sphere using the slicing method.

2) Use the slicing method to derive the formula for the volume of a cone.

3) Use the slicing method to derive the formula for the volume of a tetrahedron with side length $a.$

Volumes by Slicing

For exercises 4 - 8, draw a typical slice and find the volume using the slicing method for the given volume.

4) A pyramid with height 6 units and square base of side 2 units, as pictured here.

$This figure is a pyramid with base width of 2 and height of 6 units.$

Solution:: Here the cross-sections are squares taken perpendicular to the $y$-axis.
We use the vertical cross-section of the pyramid through its center to obtain an equation relating $x$ and $y$.
Here this would be the equation, $ y = 6 - 6x $. Since we need the dimensions of the square at each $y$-level, we solve this equation for $x$ to get, $x = 1 - \tfrac{y}{6}$.
This is half the distance across the square cross-section at the $y$-level, so the side length of the square cross-section is, $s = 2\left(1 - \tfrac{y}{6}\right).$
Thus, we have the area of a cross-section is,

$A(y) = \left[2\left(1 - \tfrac{y}{6}\right)\right]^2 = 4\left(1 - \tfrac{y}{6}\right)^2.$

$\begin{align*} \text{Then},\quad V &= \int_0^6 4\left(1 - \tfrac{y}{6}\right)^2 \, dy \\[5pt]
&= -24 \int_1^0 u^2 \, du, \quad \text{where} \, u = 1 - \tfrac{y}{6}, \, \text{so} \, du = -\tfrac{1}{6}\,dy, \quad \implies \quad -6\,du = dy \\[5pt]
&= 24 \int_0^1 u^2 \, du = 24\dfrac{u^3}{3}\bigg|_0^1 \\[5pt]
&= 8u^3\bigg|_0^1 \\[5pt]
&= 8\left( 1^3 - 0^3 \right) \quad= \quad 8\, \text{units}^3 \end{align*}$

5) A pyramid with height 4 units and a rectangular base with length 2 units and width 3 units, as pictured here.

$This figure is a pyramid with base width of 2, length of 3, and height of 4 units.$

6) A tetrahedron with a base side of 4 units,as seen here.

$This figure is an equilateral triangle with side length of 4 units.$

Answer: $V = \frac{32}{3\sqrt{2}} = \frac{16\sqrt{2}}{3}$ units³

7) A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here.

$This figure is a pyramid with a triangular base. The view is of the base. The sides of the triangle measure 6 units, 8 units, and 8 units. The height of the pyramid is 5 units.$

8) A cone of radius $ r$ and height $ h$ has a smaller cone of radius $ r/2$ and height $ h/2$ removed from the top, as seen here. The resulting solid is called a frustum.

$This figure is a 3-dimensional graph of an upside down cone. The cone is inside of a rectangular prism that represents the xyz coordinate system. the radius of the bottom of the cone is “r” and the radius of the top of the cone is labeled “r/2”.$

Answer: $V = \frac{7\pi}{12} hr^2$ units³

For exercises 9 - 14, draw an outline of the solid and find the volume using the slicing method.

9) The base is a circle of radius $ a$. The slices perpendicular to the base are squares.

10) The base is a triangle with vertices $ (0,0),(1,0),$ and $ (0,1)$. Slices perpendicular to the $xy$-plane are semicircles.

Answer

$This figure shows the x-axis and the y-axis with a line starting on the x-axis at (1,0) and ending on the y-axis at (0,1). Perpendicular to the xy-plane are 4 shaded semi-circles with their diameters beginning on the x-axis and ending on the line, decreasing in size away from the origin.$

$ V = \int_0^1 \frac{\pi(1-x)^2}{8}\, dx \quad = \quad \frac{ \pi }{24}$ units³

11) The base is the region under the parabola $ y=1−x^2$ in the first quadrant. Slices perpendicular to the $xy$-plane are squares.

12) The base is the region under the parabola $ y=1−x^2$ and above the $x$-axis. Slices perpendicular to the $y$-axis are squares.

Answer

$This figure shows the x-axis and the y-axis in 3-dimensional perspective. On the graph above the x-axis is a parabola, which has its vertex at y=1 and x-intercepts at (-1,0) and (1,0). There are 3 square shaded regions perpendicular to the x y plane, which touch the parabola on either side, decreasing in size away from the origin.$

$ V = \int_0^1 4(1 - y)\,dy \quad = \quad 2$ units³

13) The base is the region enclosed by $ y=x^2$ and $ y=9.$ Slices perpendicular to the $x$-axis are right isosceles triangles.

14) The base is the area between $ y=x$ and $ y=x^2$. Slices perpendicular to the $x$-axis are semicircles.

Answer

$This figure is a graph with the x and y axes diagonal to show 3-dimensional perspective. On the first quadrant of the graph are the curves y=x, a line, and y=x^2, a parabola. They intersect at the origin and at (1,1). Several semicircular-shaped shaded regions are perpendicular to the x y plane, which go from the parabola to the line and perpendicular to the line.$

$ V = \int_0^1 \frac{\pi}{8}\left( x - x^2 \right)^2 \, dx \quad=\quad \frac{ \pi }{240}$ units³

For exercises 15 - 16, find the volume of the solid described.

15) The base is the region between $ y=x$ and $ y=x^2$. Slices perpendicular to the $x$-axis are semicircles.

16) The base is the region enclosed by the generic ellipse $ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1.$ Slices perpendicular to the $x$-axis are semicircles.

Answer: $V = \frac{2ab^2 \pi }{3}$ units³

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

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