# How do you find the volume of region bounded by graphs of #y = x^2# and #y = sqrt x# about the x-axis?

From the graph we can see that the volume we seek is between the two functions. In order to find this, we must find the volume of revolution of

First we need to find the upper and lower bounds. We know the lower bound is

Squaring:

Volume of

Plugging in upper and lower bounds:

Volume of

Plugging in upper and lower bounds:

Required volume is:

Volume of revolution:

By signing up, you agree to our Terms of Service and Privacy Policy

To find the volume of the region bounded by the graphs of (y = x^2) and (y = \sqrt{x}) about the x-axis, you would integrate the difference between the two functions squared, from the x-coordinate of their intersection to the x-coordinate where they end or begin. The integral formula is:

[V = \pi \int_{a}^{b} \left((f(x))^2 - (g(x))^2\right) , dx]

where: (f(x)) is the upper function, (g(x)) is the lower function, (a) and (b) are the x-coordinates of the intersection points.

In this case: (f(x) = x^2) (upper function) (g(x) = \sqrt{x}) (lower function)

First, you need to find the intersection points by setting the two equations equal to each other and solving for (x). Then, integrate the difference between (f(x)) and (g(x)) squared from the lower x-coordinate to the upper x-coordinate of the intersection points. Finally, multiply the result by (\pi) to get the volume.

By signing up, you agree to our Terms of Service and Privacy Policy

- How do you find the volume bounded by #y = x^3#, y = -x, and y =1 revolved about the x=1?
- How do you find the particular solution to #sqrtx+sqrtyy'=0# that satisfies y(1)=4?
- How do you find the arc length of the curve #y = 4x^(3/2) - 1# from [4,9]?
- How do you solve the differential equation #y'=e^(-y)(2x-4)#, where #y5)=0# ?
- How do you find the volume of the region enclosed by the curves #y=2x#, #y=x^2# rotated about the x-axis?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7