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

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

32 3 2 units 3

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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.
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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”.

7 π 12 h r 2 units 3

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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.

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


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.
π 24 units 3

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

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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.


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.
2 units 3

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

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


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.
π 240 units 3

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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 , and y = 0

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y = 2 x 2 , x = 0 , x = 4 , and y = 0


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.
4096 π 5 units 3

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y = e x + 1 , x = 0 , x = 1 , and y = 0

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y = x 4 , x = 0 , and y = 1


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.
8 π 9 units 3

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y = x , x = 0 , x = 4 , and y = 0

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y = sin x , y = cos x , and x = 0


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.
π 2 units 3

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y = 1 x , x = 2 , and y = 3

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x 2 y 2 = 9 and x + y = 9 , y = 0 and x = 0


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.
207 π units 3

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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 1 2 x , x = 0 , and y = 0

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y = 2 x 3 , x = 0 , x = 1 , and y = 0


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.
4 π 5 units 3

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y = 3 x 2 , x = 0 , and y = 3

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y = 4 x 2 , y = 0 , and x = 0


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.
16 π 3 units 3

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y = 1 x + 1 , x = 0 , and x = 3

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x = sec ( y ) and y = π 4 , y = 0 and x = 0


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.
π units 3

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y = 1 x + 1 , x = 0 , and x = 2

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y = 4 x , y = x , and x = 0


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.
16 π 3 units 3

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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 , and x = 5

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y = x 2 and y = x + 2


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.
72 π 5 units 3

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y = 4 x 2 and y = 2 x


This figure is a shaded region bounded above by the curve y=4-x^2 and below by the line y=2-x.
108 π 5 units 3

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[T] y = cos x , y = e x , x = 0 , and x = 1.2927

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y = x and y = x 2


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.
3 π 10 units 3

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y = sin x , y = 5 sin x , x = 0 and x = π

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y = 1 + x 2 and y = 4 x 2


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).
2 6 π units 3

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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 = x , x = 4 , and y = 0

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y = x + 2 , y = 2 x 1 , and x = 0


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.
9 π units 3

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x = e 2 y , x = y 2 , y = 0 , and y = ln ( 2 )


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).
π 20 ( 75 4 ln 5 ( 2 ) ) units 3

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x = 9 y 2 , x = e y , y = 0 , and y = 3

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Yogurt containers can be shaped like frustums. Rotate the line y = 1 m 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.

m 2 π 3 ( b 3 a 3 ) units 3

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Rotate the ellipse ( x 2 / a 2 ) + ( 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.
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Rotate the ellipse ( x 2 / a 2 ) + ( y 2 / b 2 ) = 1 around the y -axis to approximate the volume of a football.

4 a 2 b π 3 units 3

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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.
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What is the volume of the Bundt cake that comes from rotating y = sin x around the y -axis from x = 0 to x = π ?

This figure is a graph of a 3-dimensional solid. It is round, bigger towards the bottom. It has a hole in the center that progressively gets smaller towards the bottom. Next to the graph is an image of a bundt cake, resembling the solid.

2 π 2 units 3

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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.

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The base is the region enclosed by the generic ellipse ( x 2 / a 2 ) + ( y 2 / b 2 ) = 1 . Slices perpendicular to the x -axis are semicircles.

2 a b 2 π 3 units 3

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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”.
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Find the volume common to two spheres of radius r with centers that are 2 h 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.

π 12 ( r + h ) 2 ( 6 r h ) units 3

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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”.
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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”.

π 3 ( h + R ) ( h 2 R ) 2 units 3

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Questions & Answers

I don't understand the formula
Adaeze Reply
who's formula
funny
What is a independent variable
Sifiso Reply
a variable that does not depend on another.
Andrew
solve number one step by step
bil Reply
x-xcosx/sinsq.3x
Hasnain
x-xcosx/sin^23x
Hasnain
how to prove 1-sinx/cos x= cos x/-1+sin x?
Rochel Reply
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?
Rochel Reply
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?
Rochel Reply
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
not a bad question
Salim
what is function.
Nawaz Reply
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
vector Reply
find volume of solid about y axis and y=x^3, x=0,y=1
amisha Reply
3 pi/5
vector
what is the power rule
Vanessa Reply
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?
Itumeleng Reply
Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
Cabdalla Reply
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
mahin Reply
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)
riyad Reply
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
Practice Key Terms 5

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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