The base of a solid is the region bounded by the lines y = 5x, y = 10, and x = 0. Answer the following.
a) Find the volume if the solid has cross sections perpendicular to the y-axis that are semicircles.
b) Find the volume if the solid has cross sections perpendicular to the x-axis that are semicircles.
I'm confused as to how to do this. Could you please help? Thanks!
(a) The semicircles have a diameter equal to x, or y/5. So, each semicircle has an area of
πd^2/2 = π/2 (y/5)^2 = πy^2/50
Now add up all the thin discs and you get a volume of
∫[0,10] π/50 y^2 dy = 20π/3
(b) Now the sections have diameter equal to y = 5x, so their area is
π/2 (5x)^2 = 25π/2 x^2
and thus the volume is
∫[0,2] 25π/2 x^2 dx = 100π/3
Thank you so much!
Of course! To find the volume of a solid with given cross sections, we need to integrate the areas of those cross sections.
a) To find the volume when the cross sections are semicircles perpendicular to the y-axis, we'll integrate the areas of the semicircles with respect to y. Here's how you can do it step by step:
1. Sketch the diagram of the base of the solid, which is the region bounded by the lines y = 5x, y = 10, and x = 0. It will be a triangular region.
2. Find the equation for the top boundary of the triangular region by setting y = 5x to y = 10:
5x = 10
x = 2
3. Set up the integral for the volume:
∫[from y = 5x to y = 10] (area of semicircle) dy
4. Determine the radius of each semicircle at any given value of y. Since the cross sections are perpendicular to the y-axis, the radius will be determined by the x-coordinate of the point where the semicircle intersects the line y = 5x.
5. Express the radius in terms of y by rewriting the equation y = 5x as x = (1/5)y.
6. The area of a semicircle is (πr^2)/2, where r is the radius. Substituting the expression for the radius in terms of y, the area of the semicircle becomes (π/2)(1/5)^2y^2 = (π/50)y^2.
7. Finally, integrate the area function with respect to y:
Volume = ∫[from 5 to 10] (π/50)y^2 dy
b) To find the volume with cross sections that are semicircles perpendicular to the x-axis, we will integrate the areas of the semicircles with respect to x. Here are the steps:
1. Sketch the diagram of the base of the solid, which is the region bounded by the lines y = 5x, y = 10, and x = 0. It will be a triangular region.
2. Determine the range of x-values that correspond to the base of the solid. In this case, it is from x = 0 to x = 2, as determined in step 2 of part (a).
3. Set up the integral for the volume:
∫[from x = 0 to x = 2] (area of semicircle) dx
4. Find the equation for the top boundary of the triangular region. In this case, the boundary remains y = 10.
5. Determine the radius of each semicircle at any given value of x. Since the cross sections are perpendicular to the x-axis, the radius will be determined by the y-coordinate of the point where the semicircle intersects the line y = 10.
6. Express the radius in terms of x by rewriting the equation y = 10.
7. Substituting the expression for the radius in terms of x, the area of the semicircle becomes (π/2)(10 - y)^2 = (π/2)(10 - (5x))^2.
8. Finally, integrate the area function with respect to x:
Volume = ∫[from 0 to 2] (π/2)(10 - (5x))^2 dx
These integrals can then be computed to find the volume of the solid in both cases.