# How do you find the volume of the region bounded by #y = (x)^(1/2)#; #y=0# and #x = 4# rotated about the x-axis?

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

To find the volume of the region bounded by ( y = \sqrt{x} ), ( y = 0 ), and ( x = 4 ) rotated about the x-axis, you can use the method of cylindrical shells.

The formula for the volume of a solid generated by rotating a curve ( y = f(x) ) from ( x = a ) to ( x = b ) about the x-axis using cylindrical shells is:

[ V = \int_{a}^{b} 2\pi x \cdot f(x) , dx ]

In this case, ( f(x) = \sqrt{x} ), ( a = 0 ), and ( b = 4 ). Substituting these values into the formula:

[ V = \int_{0}^{4} 2\pi x \cdot \sqrt{x} , dx ]

To solve this integral, simplify ( x \cdot \sqrt{x} = x^{3/2} ), then integrate:

[ V = \int_{0}^{4} 2\pi x^{3/2} , dx ]

[ V = 2\pi \int_{0}^{4} x^{3/2} , dx ]

[ V = 2\pi \left[ \frac{2}{5}x^{5/2} \right]_{0}^{4} ]

[ V = \frac{4\pi}{5}(4)^{5/2} ]

[ V = \frac{4\pi}{5} \cdot 8\sqrt{2} ]

[ V = \frac{32\pi\sqrt{2}}{5} ]

Therefore, the volume of the region bounded by ( y = \sqrt{x} ), ( y = 0 ), and ( x = 4 ) rotated about the x-axis is ( \frac{32\pi\sqrt{2}}{5} ).

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

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is a general solution to the differential equation #dy/dt+3ty=sint#?
- Solve # (1+x^2)^2y'' + 2x(1+x^2)y'+4y = 0 #?
- What is the arc length of #f(x)=xsinx-cos^2x # on #x in [0,pi]#?
- How do you find the carrying capacity of a population growing logistically?
- How do you calculate the arc length of the curve #y=x^2# from #x=0# to #x=4#?

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