Find the volume of the solid whose base is the region in the xy-plane bounded by the given curves and whose cross-sections perpendicular to the x-axis are (a) squares, (b) semicircles, and (c) equilater triangles.
y=x^2, x=0, x=2, y=0
I know how to graph what is given, but I don't how to use the graph to do (a), (b), and (c). I have two other problems like this one. It would be great if you could show me how to do this one to do others like this one.
There are two ways to do this. Write the area of a crosssection as a function of height. dV= area dh, then integrate h from zero to y. Surely some examples are in your text, these are standard fare in a first year calc course.
I don't think I can type out meaningful help here in ASCII.
To find the volume of the solid with different types of cross-sections, one approach is to use integration. Here's how you can calculate the volume for each case:
(a) For squares:
- Start by visualizing the region bounded by the given curves in the xy-plane.
- The curve y = x^2 intersects the x-axis at x = 0 and x = 2.
- The region between x = 0 and x = 2 is a square in shape.
- To find the volume using squares as cross-sections, divide the region into infinitesimally small vertical segments with width "dx." Each segment can be thought of as a small square.
- The height of each square is given by the difference between the upper curve (y = x^2) and the lower curve (y = 0).
- The volume of each square is then given by the base area (1*d^2) multiplied by the width "dx."
- Integrate the volume expression over the interval x = 0 to x = 2 to calculate the total volume.
- The integral expression for the volume using squares is: ∫[0 to 2] (x^2 - 0)^2 dx
(b) For semicircles:
- Similar to the previous case, start by visualizing the region in the xy-plane.
- The region between x = 0 and x = 2 will have cross-sections in the shape of semicircles when rotated around the x-axis.
- Divide the region into small vertical segments with width "dx."
- Each segment represents a semicircle with a radius equal to the y-value (x^2) at that particular x-coordinate.
- Calculate the volume of each semicircle by using the formula V = (π/2) * r^2 * dx, where r is the radius of the semicircle and dx is the width of each segment.
- Integrate the volume expression over the interval x = 0 to x = 2 to find the total volume.
- The integral expression for the volume using semicircles is: ∫[0 to 2] (π/2) * (x^2)^2 * dx.
(c) For equilateral triangles:
- Once again, visualize the region in the xy-plane.
- The region between x = 0 and x = 2 can be divided into infinitesimally small vertical segments with width "dx."
- Each segment represents an equilateral triangle with a base equal to the y-value (x^2) at that particular x-coordinate.
- The height of each triangle can be found using trigonometry. Since the triangles are equilateral, the height is equal to (√3/2) times the base.
- Calculate the volume of each triangle using the formula V = (1/2) * base * height * dx.
- Integrate the volume expression over the interval x = 0 to x = 2 to obtain the total volume.
- The integral expression for the volume using equilateral triangles is: ∫[0 to 2] (1/2) * x^2 * (√3/2) * x^2 * dx.
Using the appropriate formulas and calculating the integrals will give you the volume of the solid for each cross-sectional shape. You can repeat these steps for your other similar problems.