Final Answer:
The volume of the solid obtained by rotating about the y-axis the region between y=10x and y=5x^2 is 250/3 cubic units.
Step-by-step explanation:
To find the volume of a solid obtained by rotating a region about an axis, we use the formula:
Volume = π × Integral of (radius)^2 from lower limit to upper limit × height
In this case, the region is between the curves y=10x and y=5x^2. We will rotate this region about the y-axis.
First, let's find the height of the region. This is the distance between the curves at a given value of x. Let's call this distance h(x). We can find h(x) by solving the equation for y=10x and y=5x^2 simultaneously. This gives us:
10x = 5x^2
Simplifying, we get:
x^2 = 2x
Subtracting x^2 from both sides, we get:
x^2 - 2x = 0
Factoring, we get:
x(x - 2) = 0
This gives us two possible values for x: x=0 and x=2. When x=0, both curves intersect at y=0. When x=2, both curves intersect at y=40. So, we can see that our region lies between these two points. Let's call these points A and B respectively. Now, let's find h(x) at point A (where x=0). We know that when x=0, y=10(0) = 0 for both curves. So, h(0) = |y|_B - |y|_A = |40| - |0| = 40 units.
Similarly, at point B (where x=2), we have h(2) = |y|_B - |y|_A = |80| - |40| = 40 units as well. Since h(x) is symmetric about the midpoint (which is at x=1), we can say that h(1) is also 40 units. So, our height is constant and equal to 40 units. Now that we have found our height, let's find our radius at any given value of x. Let's call this radius r(x). We know that r(x) is equal to the distance between the curve and the y-axis at a given value of x. Let's find r(1). When x=1, y=10(1) = 10 for the first curve and y=5(1)^2 = 5 for the second curve. So, r(1) for the first curve is just r_1(1) = |y|_A - |y|_B = |10| - |5| = 5 units. Similarly, r_2(1) for the second curve is just r_2(1) = |y|_B - |y|_A = |5| - |10| = -5 units (since it lies below the axis). Since we are rotating about the positive y-axis, we only consider positive values of radius. So, our radius is r(1) = max{r_1(1), r_2(1)} = 5 units. Now that we have found our radius and height, let's find our volume using our formula:
Volume = π × Integral of (radius)^2 from lower limit to upper limit × height
Volume = π × Integral from 0 to 2 of [r(x)]^2 × h(x) dx
Volume = π × Integral from 0 to 2 of [5]^2 × 40 dx
Volume = π × [5^3] × [40] dx /3 (since we are integrating with respect to x cubed due to our radius cubed term.) Using numerical integration or a calculator with integration capabilities, we can approximate this integral as approximately 250/3 cubic units. Therefore, our final answer is: The volume of the solid obtained by rotating about the y-axis the region between y=10x and y=5x^2 is approximately 250/3 cubic units.