What is the surface area of the solid created by revolving #f(x) = x^2-x , x in [2,7]# around the x axis?
See the answer below:
By signing up, you agree to our Terms of Service and Privacy Policy
To find the surface area of the solid created by revolving ( f(x) = x^2 - x ) around the x-axis over the interval ( x \in [2, 7] ), you can use the formula for the surface area of a solid of revolution:
[ A = 2\pi \int_{a}^{b} f(x) \sqrt{1 + \left(\frac{df}{dx}\right)^2} , dx ]
Where ( a = 2 ) and ( b = 7 ), and ( f(x) = x^2 - x ).
First, find ( \frac{df}{dx} ):
[ \frac{df}{dx} = \frac{d}{dx} (x^2 - x) = 2x - 1 ]
Now, plug the function and its derivative into the formula:
[ A = 2\pi \int_{2}^{7} (x^2 - x) \sqrt{1 + (2x - 1)^2} , dx ]
[ = 2\pi \int_{2}^{7} (x^2 - x) \sqrt{1 + 4x^2 - 4x + 1} , dx ]
[ = 2\pi \int_{2}^{7} (x^2 - x) \sqrt{4x^2 - 4x + 2} , dx ]
This integral may require techniques like substitution or other advanced methods to solve, as it involves a square root. Once you find the antiderivative and evaluate it over the interval ([2, 7]), you will have the surface area of the solid of revolution.
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.
- How do you find the arc length of the curve # f(x)=e^x# from [0,20]?
- How do you find the volume of the solid generated by revolving the region bounded by the graphs #y=x, y=0, y=4, x=6#, about the line x=6?
- How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #y = 1 + x^2#, #y = 0#, #x = 0#, #x = 2# rotated about the x-axis?
- How do you set up an integral for the length of the curve #y=sqrtx, 1<=x<=2#?
- How do you find all solutions of the differential equation #(d^2y)/(dx^2)=-4y#?

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