what is the volume of the solid obtained by rotating the region bounded by y=x^2, and y=5x about the line x=5 using the washer method and respect to dy?
v =∫[0,25] π(R^2-r^2) dy
where R=5-y/5 and r=5-√y
now just crank it out.
Well, solving this problem is as easy as pie... or should I say, as easy as rotating a pie? Anyway, let's dive into it.
To find the volume using the washer method, we need to express the equations in terms of y, which is totally fine.
First, let's find the intersection points of the two curves y = x^2 and y = 5x.
Setting them equal to each other, we have x^2 = 5x.
Now, this equation has two solutions, x = 0 and x = 5.
Since we're rotating around the line x = 5, we'll rotate between these two limits.
The outer radius (R) of the washer will be 5 - x, while the inner radius (r) will be 5 - √y.
To find the volume element, we'll use the formula pi * (R^2 - r^2) * dy.
Applying the formula, the volume element becomes pi * ((5 - x)^2 - (5 - √y)^2) * dy.
Substituting x = 5 - √y and simplifying, we get pi * (10√y - 2y) * dy.
To obtain the total volume, we integrate this expression from y = 0 to y = 25.
So, the volume of the solid is the integral of pi * (10√y - 2y) * dy from 0 to 25.
But hey, don't let the math scare you! Remember, I'm just a clown bot here to make you smile, not to crunch numbers.
To find the volume of the solid obtained by rotating the region bounded by y = x^2, y = 5x, and x = 5 about the line x = 5 using the washer method with respect to dy, follow these steps:
Step 1: Find the limits of integration:
Since we are rotating about the line x = 5, we need to find the limits of integration for y. To do this, set the two equations y = x^2 and y = 5x equal to each other, and solve for x:
x^2 = 5x
x^2 - 5x = 0
x(x - 5) = 0
This gives us two possible values for x: x = 0 and x = 5. However, since we are rotating the region between y = x^2 and y = 5x, we only need to consider the values of y within this range. Therefore, the limits of integration for y are from y = 0 to y = 5*5 = 25.
Step 2: Set up the integral:
To set up the integral using the washer method, we need to express the volume element as a function of y. In this case, the volume element is a thin washer with an inner radius (r1) and an outer radius (r2).
The equation of the curve y = x^2 can be rewritten as x = √y. The equation of the line y = 5x can be rewritten as x = y/5.
The inner radius (r1) is the distance from the axis of rotation (x = 5) to the curve y = x^2, which is 5 - √y.
The outer radius (r2) is the distance from the axis of rotation (x = 5) to the line y = 5x, which is 5 - (y/5).
The volume element (dV) can be expressed as π(r2^2 - r1^2)dy.
Step 3: Evaluate the integral:
The volume (V) is given by the integral of the volume element over the limits of integration:
V = ∫[from 0 to 25] π[(5 - (y/5))^2 - (5 - √y)^2] dy
This integral can be simplified and evaluated to find the volume of the solid.
Please note that the computation of this integral involves the expansion and simplification of terms, followed by integration. The final step is to substitute the limits of integration (0 and 25) into the integral and compute the result.
To find the volume of the solid obtained by rotating the region bounded by the curves y = x^2, y = 5x, and the line x = 5 about the line x = 5 using the washer method and with respect to dy, we need to follow these steps:
Step 1: Sketch the region
First, let's sketch the region to get a better understanding of the problem. The region is bound by the curves y = x^2 and y = 5x, and lies between x = 0 and x = 5.
Step 2: Determine the limits of integration
Since we are integrating with respect to y, we need to find the limits of integration in terms of y. To do this, we set the two curves equal to each other and solve for the common values of y.
x^2 = 5x
x^2 - 5x = 0
x(x - 5) = 0
The solutions are x = 0 and x = 5, so the limits of integration are y = 0 and y = 5.
Step 3: Set up the integral
The volume of a solid using the washer method and with respect to dy is given by:
V = ∫[a,b] π(R^2 - r^2) dy
where a and b are the limits of integration, R is the outer radius, and r is the inner radius.
Step 4: Determine the outer and inner radii
To determine the outer and inner radii, we need to find the distance between the line x = 5 and the curves y = x^2 and y = 5x.
From y = x^2, we can solve for x:
x^2 = y
=> x = √y
From y = 5x, we can solve for x:
5x = y
=> x = y/5
The outer radius (R) is the distance from the line x = 5 to the curve y = 5x, so R = 5 - (y/5) = 5 - y/5.
The inner radius (r) is the distance from the line x = 5 to the curve y = x^2, so r = 5 - √y.
Step 5: Evaluate the integral
Now we can set up and evaluate the integral using the washer method formula:
V = ∫[0,5] π((5 - y/5)^2 - (5 - √y)^2) dy
This integral can be simplified and then solved using basic integral techniques.
Once you evaluate the integral, you will get the volume of the solid.