The region in the x-y plane bounded by the curve y=x^2, the line x=0 and the line x=5 and the x-axis. find the volume generated.
can you help me check for correction
the answer i got was = 625
You want to find volume. Are you rotating the region around a line, such as the x axis, or using cross sections?
Thanks, it was not specified.
As understood, find volume.
You say you got 625. What method did you use to find that number?
That is an area, not volume. Now, if that area is rotated about some axis, one can generate an volume, but the axis has to be specified. As the question stands now, it is meaningless.
I did it once for you rotating about the y axis. But in reality, it could be rotated about any axis to generate a volume.
If about x axis, I got 625 pi
It is hard to not get a pi if you spin something.
To find the volume generated by rotating the region bounded by the curve y=x^2, the line x=0, the line x=5, and the x-axis, we need to determine the method of rotation.
If we rotate the region around the x-axis, we can use the method of disks or washers to find the volume.
To use the method of disks, we divide the region into infinitesimally small disks perpendicular to the x-axis. The volume of each disk is given by πr^2h, where r is the radius of the disk and h is its thickness or height.
In this case, if we rotate the region around the x-axis, the radius of each disk is equal to y (since x^2 = y) and the thickness is dx. Therefore, the volume of each disk is π(y^2)dx.
To find the total volume, we integrate π(y^2)dx over the interval from x=0 to x=5. We need to express y in terms of x, so we substitute y=x^2 into the volume equation:
V = ∫ [from 0 to 5] π(x^2)^2dx
V = ∫ [from 0 to 5] πx^4dx
Evaluating this integral, we get:
V = π * [x^5/5] [from 0 to 5]
V = π * [5^5/5] - π * [0^5/5]
V = π * 625 - 0
V = 625π
So the volume generated is 625π.