# 6.2 Determining volumes by slicing  (Page 7/12)

 Page 7 / 12

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

$\frac{32}{3\sqrt{2}}$ units 3

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

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

$\frac{7\pi }{12}h{r}^{2}$ units 3

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

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

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

$\frac{\pi }{24}$ units 3

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

The base is the region under the parabola $y=1-{x}^{2}$ and above the $x\text{-axis}\text{.}$ Slices perpendicular to the $y\text{-axis}$ are squares.

$2$ units 3

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

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

$\frac{\pi }{240}$ units 3

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

$x+y=8,x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$

$y=2{x}^{2},x=0,x=4,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$

$\frac{4096\pi }{5}$ units 3

$y={e}^{x}+1,x=0,x=1,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$

$y={x}^{4},x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=1$

$\frac{8\pi }{9}$ units 3

$y=\sqrt{x},x=0,x=4,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$

$y=\text{sin}\phantom{\rule{0.2em}{0ex}}x,y=\text{cos}\phantom{\rule{0.2em}{0ex}}x,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$

$\frac{\pi }{2}$ units 3

$y=\frac{1}{x},x=2,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3$

${x}^{2}-{y}^{2}=9\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x+y=9,y=0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$

$207\pi$ units 3

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

$y=4-\frac{1}{2}x,x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$

$y=2{x}^{3},x=0,x=1,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$

$\frac{4\pi }{5}$ units 3

$y=3{x}^{2},x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3$

$y=\sqrt{4-{x}^{2}},y=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$

$\frac{16\pi }{3}$ units 3

$y=\frac{1}{\sqrt{x+1}},x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=3$

$x=\text{sec}\left(y\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=\frac{\pi }{4},y=0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$

$\pi$ units 3

$y=\frac{1}{x+1},x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=2$

$y=4-x,y=x,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$

$\frac{16\pi }{3}$ units 3

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

$y=x+2,y=x+6,x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=5$

$y={x}^{2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=x+2$

$\frac{72\pi }{5}$ units 3

${x}^{2}={y}^{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{x}^{3}={y}^{2}$

$y=4-{x}^{2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=2-x$

$\frac{108\pi }{5}$ units 3

[T] $y=\text{cos}\phantom{\rule{0.2em}{0ex}}x,y={e}^{\text{−}x},x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=1.2927$

$y=\sqrt{x}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y={x}^{2}$

$\frac{3\pi }{10}$ units 3

$y=\text{sin}\phantom{\rule{0.2em}{0ex}}x\text{,}\phantom{\rule{0.2em}{0ex}}y=5\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}x,x=0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=\pi$

$y=\sqrt{1+{x}^{2}}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=\sqrt{4-{x}^{2}}$

$2\sqrt{6}\pi$ units 3

For the following exercises, 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.

$y=\sqrt{x},x=4,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$

$y=x+2,y=2x-1,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$

$9\pi$ units 3

$y=\sqrt[3]{x}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y={x}^{3}$

$x={e}^{2y},x={y}^{2},y=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=\text{ln}\left(2\right)$

$\frac{\pi }{20}\left(75-4\phantom{\rule{0.2em}{0ex}}{\text{ln}}^{5}\left(2\right)\right)$ units 3

$x=\sqrt{9-{y}^{2}},x={e}^{\text{−}y},y=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3$

Yogurt containers can be shaped like frustums. Rotate the line $y=\frac{1}{m}x$ around the y -axis to find the volume between $y=a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=b.$

$\frac{{m}^{2}\pi }{3}\left({b}^{3}-{a}^{3}\right)$ units 3

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

Rotate the ellipse $\left({x}^{2}\text{/}{a}^{2}\right)+\left({y}^{2}\text{/}{b}^{2}\right)=1$ around the y -axis to approximate the volume of a football.

$\frac{4{a}^{2}b\pi }{3}$ units 3

A better approximation of the volume of a football is given by the solid that comes from rotating $y=\text{sin}\phantom{\rule{0.2em}{0ex}}x$ around the x -axis from $x=0$ to $x=\pi .$ What is the volume of this football approximation, as seen here?

What is the volume of the Bundt cake that comes from rotating $y=\text{sin}\phantom{\rule{0.2em}{0ex}}x$ around the y -axis from $x=0$ to $x=\pi ?$

$2{\pi }^{2}$ units 3

For the following exercises, find the volume of the solid described.

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

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

$\frac{2a{b}^{2}\pi }{3}$ units 3

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

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

$\frac{\pi }{12}{\left(r+h\right)}^{2}\left(6r-h\right)$ units 3

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

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

$\frac{\pi }{3}\left(h+R\right){\left(h-2R\right)}^{2}$ units 3

I don't understand the formula
who's formula
funny
What is a independent variable
a variable that does not depend on another.
Andrew
solve number one step by step
x-xcosx/sinsq.3x
Hasnain
x-xcosx/sin^23x
Hasnain
how to prove 1-sinx/cos x= cos x/-1+sin x?
1-sin x/cos x= cos x/-1+sin x
Rochel
how to prove 1-sun x/cos x= cos x / -1+sin x?
Rochel
how to prove tan^2 x=csc^2 x tan^2 x-1?
divide by tan^2 x giving 1=csc^2 x -1/tan^2 x, rewrite as: 1=1/sin^2 x -cos^2 x/sin^2 x, multiply by sin^2 x giving: sin^2 x=1-cos^2x. rewrite as the familiar sin^2 x + cos^2x=1 QED
Barnabas
how to prove sin x - sin x cos^2 x=sin^3x?
sin x - sin x cos^2 x sin x (1-cos^2 x) note the identity:sin^2 x + cos^2 x = 1 thus, sin^2 x = 1 - cos^2 x now substitute this into the above: sin x (sin^2 x), now multiply, yielding: sin^3 x Q.E.D.
Andrew
take sin x common. you are left with 1-cos^2x which is sin^2x. multiply back sinx and you get sin^3x.
navin
Left side=sinx-sinx cos^2x =sinx-sinx(1+sin^2x) =sinx-sinx+sin^3x =sin^3x thats proved.
Alif
how to prove tan^2 x/tan^2 x+1= sin^2 x
Rochel
Salim
what is function.
what is polynomial
Nawaz
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Alif
a term/algebraic expression raised to a non-negative integer power and a multiple of co-efficient,,,,,, T^n where n is a non-negative,,,,, 4x^2
joe
An expression in which power of all the variables are whole number . such as 2x+3 5 is also a polynomial of degree 0 and can be written as 5x^0
Nawaz
what is hyperbolic function
find volume of solid about y axis and y=x^3, x=0,y=1
3 pi/5
vector
what is the power rule
Is a rule used to find a derivative. For example the derivative of y(x)= a(x)^n is y'(x)= a*n*x^n-1.
Timothy
how do i deal with infinity in limits?
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz