find the volume of the solid whose cross section is perpendicular to the x axis from isosceles right triangles and has a leg that run from the lines y=-x to the line y=.5x
To find the volume of the solid, we can start by visualizing the cross-section of the solid. In this case, the cross-section is a right triangle with legs running from the lines y = -x and y = 0.5x, perpendicular to the x-axis.
To proceed, we need to determine the limits of integration for the variable x, which will define the boundaries of the solid in the x-direction.
Since the legs of the right triangle run from the lines y = -x to y = 0.5x, we can set these equations equal to each other to find the intersection point:
-x = 0.5x
Simplifying, we get:
1.5x = 0
x = 0
So, the intersection point between the two lines is at x = 0.
Next, we need to find the limits of integration for x. Since the solid is formed by right triangles perpendicular to the x-axis, we want to integrate along the x-axis within the range where the solid exists.
To determine these limits, we need to find the x-values where the right triangles start and end. This can be done by setting the equations of the two lines equal to 0:
For y = -x: x = 0
For y = 0.5x: 0.5x = 0
Thus, the triangles start at x = 0 and end at x = 0.
Since the triangles have no width in the x-direction, the volume would effectively be zero units cubed.
Therefore, the volume of the solid is 0 units cubed.