How do you find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis #y=7x^2#, x =1, y =0, about the x-axis?
Volume =
By signing up, you agree to our Terms of Service and Privacy Policy
To find the volume of the solid obtained by rotating the region bounded by the curves ( y = 7x^2 ), ( x = 1 ), and ( y = 0 ) about the x-axis, you can use the disk method.
-
Determine the limits of integration by finding the points of intersection between the curves ( y = 7x^2 ) and ( x = 1 ). This occurs at ( x = 1 ).
-
Set up the integral for the volume using the formula for the disk method: [ V = \pi \int_{a}^{b} (f(x))^2 , dx ]
where ( f(x) ) represents the radius of the disks, and ( a ) and ( b ) are the limits of integration.
-
Substitute the values into the integral: [ V = \pi \int_{0}^{1} (7x^2)^2 , dx ]
-
Integrate the function: [ V = \pi \int_{0}^{1} 49x^4 , dx ]
-
Evaluate the integral: [ V = \pi \left[ \frac{49}{5}x^5 \right]_{0}^{1} ] [ V = \pi \left( \frac{49}{5} \right) ]
-
Simplify the expression: [ V = \frac{49\pi}{5} ]
So, the volume of the solid obtained by rotating the region bounded by the curves about the x-axis is ( \frac{49\pi}{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.
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 fast is the volume changing with respect to time when the radius is changing with respect to time when the radius is changing at a rate of dr/dt=1.5 feet per second and r=2 feet?
- The region under the curves #y=x^2, y=x# is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
- How do you find the volume of a solid that is enclosed by #y=3x^2# and y=2x+1 revolved about the x axis?
- How do you find the volume of the solid generated by revolving the region bounded by the graphs of the equations #y=sec(x)# , y=0, #0 <= x <= pi/3# about the line y = 5?
- How do you find #\int _ { 4} ^ { 9} \frac { x + 1} { x+ 2\sqrt { x } - 3} d x#?

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