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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

how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
v=(x,y) وu=(x,y ) ∂u/∂x* ∂x/∂u +∂v/∂x*∂x/∂v=1
log tan (x/4+x/2)
Rohan
what is the procedures in solving number 1?
Vier Reply
review of funtion role?
Md Reply
for the function f(x)={x^2-7x+104 x<=7 7x+55 x>7' does limx7 f(x) exist?
find dy÷dx (y^2+2 sec)^2=4(x+1)^2
Rana Reply
Integral of e^x/(1+e^2x)tan^-1 (e^x)
naveen Reply
why might we use the shell method instead of slicing
Madni Reply
fg[[(45)]]²+45⅓x²=100
albert Reply
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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