The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
y = −x2 + 7x − 12, y = 0; about the x-axis
y = -(x-3)(x-4)
So, using discs of thickness dx,
v = ∫[3,4] πr^2 dx
where r=y
v = ∫[3,4] π(x^2-7x+12)^2 dx = π/30
Using shells of thickness dy, and taking advantage of the symmetry,
v = 2∫[0,1/4] 2πrh dy
where r=y and h=√(1/4 - y)
v = 2∫[0,1/4] 2πy√(1/4 - y) dy = π/30
A variable force of
7x−2
pounds moves an object along a straight line when it is x feet from the origin. Calculate the work done in moving the object from
x = 1
ft to
x = 19
ft. (Round your answer to two decimal places.)
Well, let's start by finding the points of intersection between the curves. We'll set y = 0 and solve for x in the equation y = -x^2 + 7x - 12.
0 = -x^2 + 7x - 12
Rearranging the equation:
x^2 - 7x + 12 = 0
Factoring the quadratic equation:
(x - 3)(x - 4) = 0
So the curves intersect at x = 3 and x = 4.
To find the volume V of the resulting solid, we'll use the method of cylindrical shells. We'll integrate the circumference of each shell multiplied by its height.
The height of each shell is the difference between the y-coordinates of the curves, which is given by h(x) = -x^2 + 7x - 12.
The radius of each shell is the x-coordinate, which is given by r(x) = x.
The circumference of each shell is given by c(x) = 2πr(x) = 2πx.
The volume of each shell is given by V(x) = c(x) * h(x) = 2πx * (-x^2 + 7x - 12).
Now we'll integrate this expression with respect to x, from x = 3 to x = 4, to find the total volume V:
V = ∫[from 3 to 4] 2πx * (-x^2 + 7x - 12) dx
Well, you know what they say about integration... it's the perfect way to find the total volume and make mathemagical dreams come true! So let's crunch those numbers and find V. Good luck!
To find the volume of the resulting solid, we can use the method of cylindrical shells.
First, we need to find the limits of integration. To do this, we find the x-values at the points where the two curves intersect. Setting y = 0 and solving for x, we have:
0 = −x^2 + 7x − 12
Factoring the quadratic equation, we get:
0 = (x - 3)(x - 4)
So the curves intersect at x = 3 and x = 4.
Next, we need to find the height of each cylindrical shell at each value of x. The height of each cylindrical shell is the difference between the y-values of the two curves at that particular x-value. Thus, the height h is given by:
h = −x^2 + 7x − 12 - 0 = -x^2 + 7x - 12
The radius of each cylindrical shell is just the x-value.
Now, we can set up the integral to calculate the volume:
V = ∫[a,b] 2πrh dx
where a and b are the x-values where the curves intersect.
Substituting in the values for the height and radius:
V = ∫[3,4] 2πx(-x^2 + 7x - 12) dx
Simplifying the expression inside the integral:
V = ∫[3,4] -2πx^3 + 14πx^2 - 24πx dx
Integrating term by term:
V = [-0.5πx^4 + 4.67πx^3 - 12πx^2] from 3 to 4
Evaluating the integral:
V = [-0.5π(4^4) + 4.67π(4^3) - 12π(4^2)] - [-0.5π(3^4) + 4.67π(3^3) - 12π(3^2)]
V = [-16π + 74.56π - 192π] - [-9π + 41.31π - 108π]
V = 10.56π - 50.13π + 84π
V = 43.43π
Therefore, the volume of the resulting solid is 43.43π cubic units.
To find the volume of the solid formed by rotating the region bounded by the curves y = -x^2 + 7x - 12 and y = 0 around the x-axis, we can use the method of cylindrical shells.
Let's start by identifying the bounds of integration. We need to find the x-values where the curves intersect, which will determine the limits of integration.
Setting -x^2 + 7x - 12 = 0, we can factor this quadratic equation:
-(x - 3)(x - 4) = 0
This gives us two x-values: x = 3 and x = 4. Therefore, the region of interest lies between x = 3 and x = 4.
Next, we need to determine the radius and the height of each cylindrical shell to calculate its volume.
The radius (r) of each cylindrical shell will be the x-value, and the height (h) will be the corresponding y-value on the curve y = -x^2 + 7x - 12.
The volume (V) of each shell is given by the formula:
V = 2πrh.
Since we are rotating about the x-axis, the lower radius will always be 0, as the shell is formed from the curve y = 0.
Now, we can calculate the volume of each shell and sum them up to find the total volume.
V = ∫[3 to 4] (2πx)(-x^2 + 7x - 12) dx
Integrating this expression will give us the volume V. We can simplify the integrand before evaluating the integral.
V = 2π ∫[3 to 4] (-x^3 + 7x^2 - 12x) dx
Evaluating the integral will yield the final value of V, which represents the volume of the solid.
Note: Please ensure you use appropriate computational tools or software to perform the integration calculation accurately.