HallsofIvy said:Why in the world would you slice it vertically? Since it is rotated around the y-axis, slices horizontally will be disks with radius x. Each would have area [itex]\pi x^2= \pi y[/itex] and each infinitesmal disc will have volume [itex]\pi y dy[/itex]. Integrate that.
Where did you get y= x from? Your original post was:-EquinoX- said:I am sorry that's my mistake. I would slice it horizontally and take the integral from 0 to 1. And shouldn't the radius be 1-square root of y?? Because it's the intersection of y = x^2 and y = x
Every horizontal "slice" is a circle with center at the y-axis, x= 0, and the end of a radius at [itex]x= \sqrt{y}[/itex]. Of course, the area of the disk is [itex]\pi x^2= \pi y[/itex].-EquinoX- said:What is the volume of the region bounded by y = x^2, y = 1, and the y-axis rotated around the y -axis
To calculate the volume of a cube, you need to know the length, width, and height of the cube. The formula for volume is V = l x w x h, where l is the length, w is the width, and h is the height. Simply multiply these three values together to find the volume of the cube.
To calculate the volume of a cylinder, you need to know the radius and height of the cylinder. The formula for volume is V = π x r^2 x h, where π is approximately 3.14, r is the radius, and h is the height. First, square the radius, then multiply it by π, and finally multiply that value by the height to find the volume of the cylinder.
To calculate the volume of a sphere, you need to know the radius of the sphere. The formula for volume is V = 4/3 x π x r^3, where π is approximately 3.14 and r is the radius. First, cube the radius, then multiply it by π, and finally multiply that value by 4/3 to find the volume of the sphere.
To calculate the volume of a rectangular prism, you need to know the length, width, and height of the prism. The formula for volume is V = l x w x h, where l is the length, w is the width, and h is the height. Simply multiply these three values together to find the volume of the rectangular prism.
To calculate the volume of an irregular-shaped object, you can use the water displacement method. Fill a graduated cylinder with water and record the initial volume. Then, place the object in the cylinder and record the new volume. The difference between the two volumes is the volume of the object. Alternatively, you can use a formula or computer software to calculate the volume of an irregular-shaped object.