6.1E: Exercises

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Gilbert Strang & Edwin “Jed” Herman
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Reading Questions

What is a solid of revolution?
What is the geometric shape of a cross-sectional slice in the disk method?
What is the key difference between a slice in the disk method and a slice in the washer method?
When using the washer method for a region between $y=f(x)$ and $y=g(x)$ revolved around the x-axis, what is the correct integrand: $\pi(f(x)-g(x))^2$ or $\pi((f(x))^2 - (g(x))^2)$?
If a region is bounded by $x=g(y)$ and revolved around the y-axis, what is the variable of integration?
How do you determine the outer radius ($r_o$) and inner radius ($r_i$) when revolving the region bounded by $f(x)=4-x$ and the x-axis over $[0, 4]$ around the line $y = -2$?

1) Use the disk method to derive the formula for the volume of a trapezoidal cylinder.

2) Explain when you would use the disk method versus the washer method. When are they interchangeable?

Disk and Washer Method

For exercises 3 - 10, draw the region bounded by the curves. Then, use the disk or washer method to find the volume when the region is rotated around the $x$-axis.

3) $ x+y=8,\quad x=0$, and $ y=0$

4) $ y=2x^2,\quad x=0,\quad x=4,$ and $ y=0$

Answer

$This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=2x^2, below by the x-axis, and to the right by the vertical line x=4.$

$\displaystyle V = \int_0^4 4\pi x^4\, dx \quad=\quad \frac{4096π}{5}$ units³

5) $ y=e^x+1,\quad x=0,\quad x=1,$ and $ y=0$

6) $ y=x^4,\quad x=0$, and $ y=1$

Answer

$This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=1, below by the curve y=x^4, and to the left by the y-axis.$

$\displaystyle V = \int_0^1 \pi\left( 1^2 - \left( x^4\right)^2\right)\, dx = \int_0^1 \pi\left( 1 - x^8\right)\, dx \quad = \quad \frac{8π}{9}$ units³

7) $ y=\sqrt{x},\quad x=0,\quad x=4,$ and $ y=0$

8) $ y=\sin x,\quad y=\cos x,$ and $ x=0$

Answer

$This figure is a shaded region bounded above by the curve y=cos(x), below to the left by the y-axis and below to the right by y=sin(x). The shaded region is in the first quadrant.$

$\displaystyle V = \int_0^{\pi/4} \pi \left( \cos^2 x - \sin^2 x\right) \, dx = \int_0^{\pi/4} \pi \cos 2x \, dx \quad=\quad \frac{π}{2}$ units³

9) $ y=\dfrac{1}{x},\quad x=2$, and $ y=3$

10) $ x^2−y^2=9$ and $ x+y=9,\quad y=0$ and $ x=0$

Answer

$This figure is a graph in the first quadrant. It is a shaded region bounded above by the line x + y=9, below by the x-axis, to the left by the y-axis, and to the left by the curve x^2-y^2=9.$

$V = 207π$ units³

For exercises 11 - 18, draw the region bounded by the curves. Then, find the volume when the region is rotated around the $y$-axis.

11) $ y=4−\dfrac{1}{2}x,\quad x=0,$ and $ y=0$

12) $ y=2x^3,\quad x=0,\quad x=1,$ and $ y=0$

Answer

$This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=2x^3, below by the x-axis, and to the right by the line x=1.$

$V = \frac{4π}{5}$ units³

13) $ y=3x^2,\quad x=0,$ and $ y=3$

14) $ y=\sqrt{4−x^2},\quad y=0,$ and $ x=0$

Answer

$This figure is a graph in the first quadrant. It is a quarter of a circle with center at the origin and radius of 2. It is shaded on the inside.$

$V = \frac{16π}{3}$ units³

15) $ y=\dfrac{1}{\sqrt{x+1}},\quad x=0$, and $ x=3$

16) $ x=\sec(y)$ and $ y=\dfrac{π}{4},\quad y=0$ and $ x=0$

Answer

$This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=pi/4, to the right by the curve x=sec(y), below by the x-axis, and to the left by the y-axis.$

$V = π$ units³

17) $ y=\dfrac{1}{x+1},\quad x=0$, and $ x=2$

18) $ y=4−x,\quad y=x,$ and $ x=0$

Answer

$This figure is a graph in the first quadrant. It is a shaded triangle bounded above by the line y=4-x, below by the line y=x, and to the left by the y-axis.$

$V = \frac{16π}{3}$ units³

For exercises 19 - 26, draw the region bounded by the curves. Then, find the volume when the region is rotated around the $x$-axis.

19) $ y=x+2,\quad y=x+6,\quad x=0$, and $ x=5$

20) $ y=x^2$ and $ y=x+2$

Answer

$This figure is a graph above the x-axis. It is a shaded region bounded above by the line y=x+2, and below by the parabola y=x^2.$

$V = \frac{72π}{5}$ units³

21) $ x^2=y^3$ and $ x^3=y^2$

22) $ y=4−x^2$ and $ y=2−x$

Answer

$This figure is a shaded region bounded above by the curve y=4-x^2 and below by the line y=2-x.$

$V = \frac{108π}{5}$ units³

23) [Technology Required] $ y=\cos x,\quad y=e^{−x},\quad x=0$, and $ x=1.2927$

24) $ y=\sqrt{x}$ and $ y=x^2$

Answer

$This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=squareroot(x), below by the curve y=x^2.$

$V = \frac{3π}{10}$ units³

25) $ y=\sin x,\quad y=5\sin x,\quad x=0$ and $ x=π$

26) $ y=\sqrt{1+x^2}$ and $ y=\sqrt{4−x^2}$

Answer

$This figure is a shaded region bounded above by the curve y=squareroot(4-x^2) and, below by the curve y=squareroot(1+x^2).$

$V = 2\sqrt{6}π$ units³

For exercises 27 - 31, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the $y$-axis.

27) $ y=\sqrt{x},\quad x=4$, and $ y=0$

28) $ y=x+2,\quad y=2x−1$, and $ x=0$

Answer

$This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=x+2, below by the line y=2x-1, and to the left by the y-axis.$

$V = 9π$ units³

29) $ y=\dfrac{3}{x}$ and $ y=x^3$

30) $ x=e^{2y},\quad x=y^2,\quad y=0$, and $ y=\ln(2)$

Answer

$This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=ln(2), below by the x-axis, to the left by the curve x=y^2, and to the right by the curve x=e^(2y).$

$V = \dfrac{π}{20}(75−4\ln^5(2))$ units³

31) $ x=\sqrt{9−y^2},\quad x=e^{−y},\quad y=0$, and $ y=3$

32) Yogurt containers can be shaped like frustums. Rotate the line $ y=\left(\frac{1}{m}\right)x$ around the $y$-axis to find the volume between $ y=a$ and $ y=b$.

$This figure has two parts. The first part is a solid cone. The base of the cone is wider than the top. It is shown in a 3-dimensional box. Underneath the cone is an image of a yogurt container with the same shape as the figure.$

Answer: $V = \dfrac{m^2π}{3}(b^3−a^3)$ units³

33) Rotate the ellipse $ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ around the $x$-axis to approximate the volume of a football, as seen here.

$This figure has an oval that is approximately equal to the image of a football.$

34) Rotate the ellipse $ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ around the $y$-axis to approximate the volume of a football.

Answer: $V = \frac{4a^2bπ}{3}$ units³

35) A better approximation of the volume of a football is given by the solid that comes from rotating $ y=\sin x$ around the $x$-axis from $ x=0$ to $ x=π$. What is the volume of this football approximation, as seen here?

$This figure has a 3-dimensional oval shape. It is inside of a box parallel to the x axis on the bottom front edge of the box. The y-axis is vertical to the solid.$

For exercises 36 - 39, find the volume of the solid described.

36) Bore a hole of radius a down the axis of $a$ right cone and through the base of radius $b$, as seen here.

$This figure is an upside down cone. It has a radius of the top as “b”, center at “a”, and height as “b”.$

37) Find the volume common to two spheres of radius $r$ with centers that are $2h$ apart, as shown here.

$This figure has two circles that intersect. Both circles have radius “r”. There is a line segment from one center to the other. In the middle of the intersection of the circles is point “h”. It is on the line segment.$

Answer: $V = \frac{π}{12}(r+h)^2(6r−h)$ units³

38) Find the volume of a spherical cap of height $h$ and radius $r$ where $h<r$, as seen here.

$This figure a portion of a sphere. This spherical cap has radius “r” and height “h”.$

39) Find the volume of a sphere of radius $R$ with a cap of height $h$ removed from the top, as seen here.

$This figure is a sphere with a top portion removed. The radius of the sphere is “R”. The distance from the center to where the top portion is removed is “R-h”.$

Answer: $V = \dfrac{π}{3}(h+R)(h−2R)^2$ 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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Disk and Washer Method

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