Calculating Volume using Integrals

Key Questions

How do you find the volume of the solid with base region bounded by the curve $y = e^{x}$ $y=e^x$ , $y = ln 4$ $y=ln4$ , and the $y$ $y$ -axis if cross sections perpendicular to the $y$ $y$ -axis are squares?

Since its cross-sections are squares, its area can be expressed as
$A (y) = {(ln y)}^{2}$ $A(y)=(lny)^2$
So, the volume of the solid can be found by:
$V = \int_{1}^{ln 4} {(ln y)}^{2} d y$ $V=int_1^{ln4}(lny)^2dy$ .
Can you go from here? You might have to use integration by parts twice.
Please watch these videos for more details.

Wataru · · Aug 27 2014

Questions

How do you find the volume of a pyramid using integrals?
How do you find the volume of the solid with base region bounded by the curve $9 x^{2} + 4 y^{2} = 36$ $9x^2+4y^2=36$ if cross sections perpendicular to the $x$ $x$ -axis are isosceles right triangles with hypotenuse on the base?
How do you find the volume of the solid with base region bounded by the triangle with vertices $(0, 0)$ $(0,0)$ , $(1, 0)$ $(1,0)$ , and $(0, 1)$ $(0,1)$ if cross sections perpendicular to the $x$ $x$ -axis are squares?
How do you find the volume of the solid with base region bounded by the curve $y = 1 - x^{2}$ $y=1-x^2$ and the $x$ $x$ -axis if cross sections perpendicular to the $y$ $y$ -axis are squares?
How do you find the volume of the solid with base region bounded by the curve $y = 1 - x^{2}$ $y=1-x^2$ and the $x$ $x$ -axis if cross sections perpendicular to the $x$ $x$ -axis are isosceles triangles with height equal base?
How do you use an integral to find the volume of a solid torus?
How do you find the volume of the solid with base region bounded by the curves $y = 1 - x^{2}$ $y=1-x^2$ and $y = x^{2} - 9$ $y=x^2-9$ if cross sections perpendicular to the $x$ $x$ -axis are squares?
How do you find the volume of the solid with base region bounded by the curve $y = e^{x}$ $y=e^x$ , $y = ln 4$ $y=ln4$ , and the $y$ $y$ -axis if cross sections perpendicular to the $y$ $y$ -axis are squares?
How do you find the volume of a rotated region bounded by $y = \sqrt{x}$ $y=sqrt(x)$ , $y = 3$ $y=3$ , the y-axis about the y-axis?
How do you find the volume of the parallelepiped determined by the vectors: <1,3,7>, <2,1,5> and <3,1,1>?
How do you find the volume of the parallelepiped with adjacent edges pq, pr, and ps where p(3,0,1), q(-1,2,5), r(5,1,-1) and s(0,4,2)?
How do you find the volume of the wedge-shaped region on the figure contained in the cylinder $x^{2} + y^{2} = 16$ $x^2 + y^2 = 16$ and bounded above by the plane $z = x$ $z = x$ and below by the xy-plane?
How do you find the volume of the solid that lies within the sphere $x^{2} + y^{2} + z^{2} = 25$ $x^2+y^2+z^2 =25$ , above the xy plane, and outside the cone?
What is the volume of the solid generated when S is revolved about the line $y = 3$ $y=3$ where S is the region enclosed by the graphs of $y = 2 x$ $y=2x$ and $y = 2 x^{2}$ $y=2x^2$ and x is between [0,1]?
What is the volume of the solid given the base of a solid is the region in the first quadrant bounded by the graph of #y=-x^2+5x-4^ and the x-axis and the cross-sections of the solid perpendicular to the x-axis are equilateral triangles?
How do you find the volume of the region bounded by the graph of $y = x^{2} + 1$ $y = x^2+1$ for x is [1,2] rotated around the x axis?
How do you find the volume of the resulting solid by any method of $x^{2} + {(y - 1)}^{2} = 1$ $x^2+(y-1)^2=1$ about the x-axis?
How do you find the area enclosed by $y = sin x$ $y=sin x$ and the x-axis for $0 \leq x \leq π$ $0≤x≤pi$ and the volume of the solid of revolution, when this area is rotated about the x axis?
How do you find the volume of the solid formed by rotating the region enclosed by ?
Let R be the region between the graphs of $y = 1$ $y=1$ and $y = sin x$ $y=sinx$ from x=0 to x=pi/2, how do you find the volume of region R revolved about the x-axis?
Let R be the region in the first quadrant enclosed by the graphs of $y = e^{- x^{2}}$ $y=e^(-x^2)$ , $y = 1 - cos x$ $y=1-cosx$ , and the y axis, how do you find the volume of the solid generated when the region R is revolved about the x axis?
Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of $y = sin (x)$ $y=sin(x)$ and $y = cos (x)$ $y=cos(x)$ , how do you find the volume of the solid generated when R is revolved about the x-axis?
Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of $y = sin (x)$ $y=sin(x)$ and $y = cos (x)$ $y=cos(x)$ , how do you find the volume of the solid whose base is R and whose cross sections, cut by planes perpendicular to the x-axis, are squares?
Let R be the region in the first quadrant enclosed by the lines $x = ln 3$ $x=ln 3$ and $y = 1$ $y=1$ and the graph of $y = e^{\frac{x}{2}}$ $y=e^(x/2)$ , how do you find the volume of the solid generated when R is revolved about the line y=-1?
Let R be the region in the first quadrant enclosed by the graph of $y = 2 cos x$ $y=2cosx$ , the x-axis, and the y-axis, how do you find the volume of the solid obtained by revolving region R about the x-axis?
The region in the first quadrant enclosed by the graphs of $y = x$ $y=x$ and $y = 2 sin x$ $y=2sinx$ is revolved about the x-axis, how do you find the volume of the solid generated?
The base of a solid is the region in the first quadrant enclosed by the graph of $y = 2 - (x^{2})$ $y= 2-(x^2)$ and the coordinate axes. If every cross section of the solid perpendicular to the y-axis is a square, how do you find the volume of the solid?
Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of $y = 4 - x^{2}$ $y=4-x^2$ and $y = 1 + 2 sin x$ $y=1+2sinx$ , how do you find the volume of the solid whose base is R and whose cross sections perpendicular to the x-axis are squares?
The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and he line x+2y=8. If the cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?
Let R be the region in the first quadrant bounded by the graph of $y = 8 - x^{\frac{3}{2}}$ $y=8-x^(3/2)$ , the x-axis, and the y-axis. What is the best approximation of the volume of the solid generated when R is revolved about the x-axis?
How do you find the volume of the solid generated by revolving the region enclosed by the parabola $y^{2} = 4 x$ $y^2=4x$ and the line y=x revolved about the x-axis?
How do you find the volume of the solid generated by revolving the region enclosed by the parabola $y^{2} = 4 x$ $y^2=4x$ and the line y=x revolved about the y-axis?
Let R be the region in the first and second quadrants bounded above by the graph of $y = \frac{20}{1 + x^{2}}$ $y=20/(1+x^2)$ and below by the horizontal line y=2, how do you find volume of the solid generated when R is rotated about the x-axis?
How do you find the volume of the solid formed when the area in the first quadrant bounded by the curves $y = e^{x}$ $y=e^x$ and x = 3?
Let R be the region in the first quadrant enclosed by the hyperbola $x^{2} - y^{2} = 9$ $x^2 -y^2= 9$ , the x-axis , the line x=5, how do you find the volume of the solid generated by revolving R about the x-axis?
How do you find the volume generated by revolving about the x-axis, the first quadrant region enclosed by the graphs of $y = 9 - x^{2}$ $y = 9 - x^2$ and $y = 9 - 3 x$ $y = 9 - 3x$ between 0 to 3?
How do you use the Disk method to set up the integral to find the volume of the solid generated by revolving about the y-axis the region bounded by the graphs of and the line $y = x$ $y = x$ , and $y = x^{3}$ $y = x^3$ between x = 0 and x = 1?
The first quadrant region enclosed by y=2x, the x-axis and the line x=1 is resolved about the line y=0. How do you find the resulting volume?
The base of a solid region in the first quadrant is bounded by the x-axis,y-axis, the graph of $y = x^{2} + 1$ $y=x^2+1$ , and the vertical line x=2. If the cross sections perpendicular to the x-axis are squares, what is the volume of the solid?
The base of a solid is the region in the first quadrant bounded by the line x+2y=4 and the coordinate axes, what is the volume of the solid if every cross section perpendicular to the x-axis is a semicircle?
How do you find the volume of the solid enclosed by the surface z=xsec^2(y) and the planes z=0, x=0,x=2,y=0, and y=π/4?
How do you find the volume V of the described solid S where the base of S is a circular disk with radius 4r and Parallel cross-sections perpendicular to the base are squares?
A solid has a circular base of radius 1. It has parallel cross-sections perpendicular to the base which are equilateral triangles. How do you find the volume of the solid?
How do you find the volume of the solid obtained by rotating the region bounded by y=5x^2 ,x=1 , and y=0, about the x-axis?
How do you compute the volume of the solid formed by revolving the fourth quadrant region bounded by $y = x^{2} - 1$ $y = x^2 - 1$ , y = 0, and x = 0 about the line y = 4?
The base of a certain solid is the triangle with vertices at (-8,4), (4,4), and the origin. Cross-sections perpendicular to the y-axis are squares. How do you find the volume of the solid?
How do you find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y=x^2, y=1, about y=2?
Let R be the region enclosed by the graphs of $y = {(64 x)}^{\frac{1}{4}}$ $y=(64x)^(1/4)$ and $y = x$ $y=x$ . How do you find the volume of the solid generated when region R is revolved about the x-axis?
How do you find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis $y = 7 x^{2}$ $y=7x^2$ , x =1, y =0, about the x-axis?
How do you find the volume of the solid generated when the regions bounded by the graphs of the given equations $y = \sqrt[3]{x}$ $y = root3x$ , x = 0, x = 8 and the x-axis are rotated about the x-axis?
How do you find the volume of the solid generated when the regions bounded by the graphs of the given equations $y = e^{- x}$ $y=e^-x$ , x= -1, x = 2 and the x-axis are rotated about the x-axis?
How do you find the volume of the solid generated when the regions bounded by the graphs of the given equations $y = \sqrt{3 - x^{2}}$ $y = sqrt (3 - x^2)$ , x=0, x=1 and the x-axis are rotated about the x-axis?
How do you find the volume of the solid generated when the regions bounded by the graphs of the given equations $y = \frac{2}{\sqrt{x}}, x = 1, x = 5$ $y = 2/sqrtx, x=1, x=5$ and the $x$ $x$ -axis are rotated about the $x$ $x$ -axis?
How do you find the volume of a solid y=x^2 and x=y^2 about the axis x=–8?
How do you find the volume of the solid whose base is the region bounded by y = x^2, y =x, x = 2 and x = 3, where cross-sections perpendicular to the x-axis are squares?
How do you find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = \sqrt{x}$ $y=sqrtx$ , y=0, and x=4 about the y-axis?
How do you find the volume of a solid obtained by revolving the graph of $y = 9 x \cdot \sqrt{16 - x^{2}}$ $y=9x*sqrt(16-x^2)$ over [0,16] about the y-axis?
How do you find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = sec (x)$ $y=sec(x)$ , y=0, $0 \leq x \leq \frac{π}{3}$ $0 <= x <= pi/3$ about the line y = 5?
Let $b > a > 0$ $b > a > 0$ be constants. Find the area of the surface generated by revolving the circle ${(x - b)}^{2} + y^{2} = a^{2}$ $(x − b)^2 + y^2 = a^2$ about the y-axis?
How do you find the volume of the solid bounded by the coordinate planes and the plane 6x + 5y + z = 6?
How do you find the volume of the solid bounded by the coordinate planes and the plane $7 x + y + z = 4$ $7x+y+z=4$ ?
How do you find the volume of the solid bounded by the coordinate planes and the plane $2 x + y + z = 3$ $2x+y+z=3$ ?
How do you find the volume of the solid bounded by the coordinate planes and the plane $8 x + 6 y + z = 6$ $8x + 6y + z = 6$ ?
How do you find the volume of the solid bounded by the coordinate planes and the plane $3 x + 2 y + z = 1$ $3x+2y+z=1$ ?
How do you find the volume of the solid bounded by the coordinate planes and the plane $5 x + 5 y + z = 6$ $5x + 5y + z = 6$ ?
How do you Find the volume of the solid that lies in the first octant and is bounded by the three coordinate planes and another plane passing through (3,0,0), (0,4,0), and (0,0,5)?
How do you find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes, and one vertex in the plane x+7y+11z=77?
How do you find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x+5y+8z=40?
The tetrahedron enclosed by the coordinates planes and the plane 2x+y+z=4, how do you find the volume?
How do you find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 4?
How do you use the triple integral to find the volume of the solid bounded by the surface $z = \sqrt{y}$ $z=sqrt y$ and the planes x+y=1, x=0, z=0?
How do you use the triple integral to find the volume of the solid in the first octant bounded by the coordinate planes and the plane 3x+6y+4z=12?
A rectangular box has three of its faces on the coordinate planes and one vertex in the first octant of the paraboloid $z = 4 - x^{2} - y^{2}$ $z = 4-x^2-y^2$ , what is the box's maximum volume?
How do you find the volume of the solid tetrahedron bounded by the coordinate planes and the plane x + 3y + z = 6?
How do you find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane $x + 6 y + 10 z = 60$ $x + 6y + 10z = 60$ ?
How do you use a triple integral to find the volume of the given the tetrahedron enclosed by the coordinate planes 2x+y+z=3?
How do you find the volume of the pyramid bounded by the plane 2x+3y+z=6 and the coordinate plane?
How do you find the volume of the solid in the first octant, which is bounded by the coordinate planes, the cylinder $x^{2} + y^{2} = 9$ $x^2+y^2=9$ , and the plane x+z=9?
How do you find the volume of the solid bounded by the coordinate planes and the plane 3x + 2y + z = 6?
How do you use Lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the given plane x + 8y + 5z = 24?
How would you find the volume bounded by the coordinate planes and by the plane 3x + 2y + 2z = 6?
How would you find the volume of the tetrahedron T bounded by the coordinate planes and the plane 3x+4y+z=10?
How do you find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 5?
How do you use Lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the given plane $x + 8 y + 7 z = 24$ $x + 8y + 7z = 24$ ?
How do you find the volume of the solid bounded by Z = 1 – y^2, x + y = 1, and the three coordinate plane?
How do you find the volume of the solid bounded by x^2+y^2=4 and z=x+y in the first octant?
How do you find the volume of the central part of the unit sphere that is bounded by the planes $x = \pm \frac{1}{5}, y = \pm \frac{1}{5} and z = \pm \frac{1}{5}$ $x=+-1/5, y=+-1/5 and z=+-1/5$ ?
How do you find the volume of a solid that is enclosed by $y = - x^{2} + 1$ $y=-x^2+1$ and $y = 0$ $y=0$ revolved about the x-axis?
How do you find the volume of a solid that is enclosed by $y = 2 x + 2$ $y=2x+2$ and $y = x^{2} + 2$ $y=x^2+2$ revolved about the x-axis?
How do you find the volume of a solid that is enclosed by $y = x^{2} - 2$ $y=x^2-2$ , $y = - 2$ $y=-2$ , and $x = 2$ $x=2$ revolved about y=-2?
How do you find the volume of a solid that is enclosed by $y = x^{2}$ $y=x^2$ , $y=0$ , and $x=2$ revolved about the x axis?
How do you find the volume of a solid that is enclosed by $y=secx$ , $x=pi/4$ , and the axis revolved about the x axis?
How do you find the volume of a solid that is enclosed by $y=x+1$ , $y=x^3+1$ , x=0 and y=0 revolved about the x axis?
How do you find the volume of a solid that is enclosed by $y=sqrt(4+x)$ , x=0 and y=0 revolved about the x axis?
How do you find the volume of a solid that is enclosed by $y=1/x$ , x=1, x=3, y=0 revolved about the y axis?
How do you find the volume of a solid that is enclosed by $y=3x^2$ and y=2x+1 revolved about the x axis?
Given ${r,s,u,v} in RR^4$ Prove that $min {r-s^2,s-u^2,u-v^2,v-r^2} le 1/4$ ?
Given $f(x, y)=x^2+y^2-2x$ , how do you the volume of the solid bounded by $z=(f(x, y)+f(y,x))/2-5/2, z = +-3?$
How do you compute the volume of the solid formed by revolving the given the region bounded by $y=sqrtx, y=2, x=0$ revolved about (a) the y-axis; (b) x=4?
Question #eb959
The region under the curve $y=sqrtx$ bounded by $0<=x<=4$ is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curve $y=sqrt(2x-4)$ bounded by $2<=x<=4$ is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curve $y=x$ bounded by $1<=x<=2$ is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curve $y=sqrt(1+x^2)$ bounded by $0<=x<=1$ is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curve $y=sqrt(x^2-4)$ bounded by $2<=x<=4$ is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curve $y=1/x$ bounded by $1<=x<=2$ is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $x, y>=0, y=x^2sqrt(1-x^4)$ is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $y=x^2, y=x$ is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $y=x^3, y=x^2$ is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $x=0, x=y-y^4$ is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $y=3/4x, y=1-x, y=x-1/x$ is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $y=cosxsqrtsinx, 0<=x<=pi/2$ is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $y=cosx-sinx, 0<=x<=pi/4$ is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $y=e^(1-2x), 0<=x<=2$ is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $y=sqrt(e^x+1), 0<=x<=3$ is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $y=1/sqrtx, 1<=x<=2$ is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $y=sqrt((2x)/(x+1)), 0<=x<=1$ is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?
The region under the curves $y=sinx/x, pi/2<=x<=pi$ is rotated about the x axis. How do you find the volumes of the two solids of revolution?
The region under the curves $y=xe^(x^3), 1<=x<=2$ is rotated about the x axis. How do you find the volumes of the two solids of revolution?
The region under the curves $y=x^-2, 1<=x<=2$ is rotated about the x axis. How do you find the volumes of the two solids of revolution?
The region under the curves $1/(x^2+1)=y, 0<=x<=2$ is rotated about the x axis. How do you find the volumes of the two solids of revolution?
The region under the curve $y=lnx/x^2, 1<=x<=2$ is rotated about the x axis. How do you find the volume of the solid of revolution?
The region is bounded by the given curves $y=0, y=sqrt(4-x^2), 0<=x<=1$ is roated about the x-axis, how do you find the volume of the two solids of revolution?
The region is bounded by the given curves $y=x, y=4-x, 0<=x<=2$ is rotated about the x-axis, how do you find the volume of the two solids of revolution?
Calculate the volume of a solid whose base is the ellipse $4x^2 + y^2 = 4$ and has vertical cross sections that are square?
Derive the formula for the volume of a sphere?
How fast is the volume changing with respect to time when the radius is changing with respect to time when the radius is changing at a rate of dr/dt=1.5 feet per second and r=2 feet?

Using Integrals to Find Areas and Volumes

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Calculating Volume using Integrals

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Using Integrals to Find Areas and Volumes