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?

dx pi y^2 from x = 0 to 5

pi dx (x^4 )
pi x^5/5
pi (625)
so I get 625 pi if we are spinning around the x axis

To find the volume generated by the region bounded by the curve y=x^2, the line x=0, the line x=5, and the x-axis, you can use the method of cylindrical shells.

First, let's visualize the region in the x-y plane. It is a parabolic shape bounded by the curve y=x^2, the x-axis, and the vertical lines x=0 and x=5.

To find the volume, we will consider an infinitely thin vertical strip at a specific x-coordinate. We will assume the width of this strip is dx.

The height of this strip will be the difference between the curve y=x^2 and the x-axis, which is y.

The length of the strip will be the difference between x=5 (the right boundary) and x (the current x-coordinate).

Now we can calculate the volume contributed by each strip:

dV = 2πxy*dx

To find the total volume, we need to integrate this expression with respect to x from 0 to 5:

V = ∫(0 to 5) dV = ∫(0 to 5) 2πxy*dx

Now, let's calculate the integral:

V = 2π∫(0 to 5) x(x^2)dx
= 2π∫(0 to 5) x^3dx
= 2π[(x^4)/4] (0 to 5)
= 2π[(5^4)/4]
= 2π[625/4]
= 125π/2

So, the correct answer is V = 125π/2, which is approximately equal to 196.35. Hence, it seems that the answer you got, 625, is not correct.