What is the surface area produced by rotating #f(x)=x^3-x^2+1, x in [0,3]# around the x-axis?
graph{(x^3-x^2+1-y)(x-3+.001y)y(x-.001y)=0 [0, 4, -1, 20]}
#(140.529)(3.1416)=441.5 cubic units, nearly.
By signing up, you agree to our Terms of Service and Privacy Policy
To find the surface area produced by rotating the function ( f(x) = x^3 - x^2 + 1 ) around the x-axis over the interval ([0,3]), we use the formula for surface area of revolution:
[ A = \int_{a}^{b} 2\pi f(x) \sqrt{1 + (f'(x))^2} , dx ]
Where ( f'(x) ) represents the derivative of ( f(x) ).
First, we find ( f'(x) ) which is ( f'(x) = 3x^2 - 2x ).
Now, we plug ( f(x) ) and ( f'(x) ) into the formula:
[ A = \int_{0}^{3} 2\pi (x^3 - x^2 + 1) \sqrt{1 + (3x^2 - 2x)^2} , dx ]
[ A = \int_{0}^{3} 2\pi (x^3 - x^2 + 1) \sqrt{1 + (9x^4 - 12x^3 + 4x^2)} , dx ]
Now, compute the integral numerically.
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.
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.
- Are the particular integral and complementary function solutions of a Differential Equation linearly independent?
- What is the volume of the solid produced by revolving #f(x)=cscx-cotx, x in [pi/8,pi/3] #around the x-axis?
- How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y = 32 - x^2# and #y= x^2# revolved about the x=4?
- How do you find the total force exerted on the five exposed faces of a cube that is on a side lying on the bottom of a swimming pool that is 20ft by 15ft by 10ft deep filled with water?
- How do you find the particular solution to #2xy'-lnx^2=0# that satisfies y(1)=2?

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