Find Volume of Solid with Trapezoid Cross Sections

[Hirokukiro]

A solid has as its base the region bounded by the curves y=-2x^2=2 and y=-x^2 +1. Find the volume of the solid if every cross section of a plane perpendicular to the x-axis is a trapezoid with lower base in the xy-plane upper base equal to 1/2 the length of the lower base, and height equal to 2 times the length of lower base.

Lower base: this is the difference between the two functions, or (-2x^2+1) -
(-x^2+1). Simplify that and you have your lower base.

Upper base: as given, it's half of the lower base. So once you know the
lower base, cut it in half and you have your upper base.

Height: This, too is given in terms of your lower base, so double that lower
base and now you have your height.

When you put it together, remember that A = (1/2)(b1+b2)(h). Plug in what
you found above and have at it!

 

[jvicens]

Thank you for your question. To find the volume of the solid described, we will need to use the method of cross-sectional area. This method involves finding the area of each cross section of the solid perpendicular to the x-axis and then adding up all the areas to get the total volume.

First, let's find the lower base of the trapezoid. As you correctly stated, the lower base is the difference between the two functions, y=-2x^2+2 and y=-x^2+1. Simplifying this, we get y=-x^2+1. This will be the lower base for each cross section.

Next, we can find the upper base by cutting the lower base in half. So the upper base will be 1/2 of the length of the lower base, or 1/2(-x^2+1). Simplifying this, we get y=-1/2x^2+1/2.

Finally, we can find the height of the trapezoid. As given in the problem, the height is equal to 2 times the length of the lower base. So the height is 2(-x^2+1), or -2x^2+2.

Now, we can use the formula for the area of a trapezoid, A=(1/2)(b1+b2)(h), to find the area of each cross section. Plugging in the values we found above, we get A=(1/2)(-x^2+1+(-1/2x^2+1/2))(-2x^2+2). Simplifying this, we get A=(-3/4x^4+5/4x^2+1)(-2x^2+2).

To find the total volume, we need to integrate this expression from the lower limit of x=0 to the upper limit of x=1. This will give us the sum of the areas of all the cross sections. So the volume of the solid is V=∫0^1(-3/4x^4+5/4x^2+1)(-2x^2+2) dx.

Evaluating this integral, we get V=1.5 units^3. Therefore, the volume of the solid bounded by the curves y=-2x^2+2 and y=-x^2