Find the volume of a solid using known cross sections?

[LilTaru]

Homework Statement

The base of a solid is the region between the parabolas x = y² and x = 3 - 2y². Find the volume of the solid given that the cross sections perpendicular to the x-axis are:

a) rectangles of height h

b) equilateral triangles

c) isosceles right triangles, hypotenuse on the xy-plane

Homework Equations

I know the volume of solid using cross sections is V = $$\int$$A(x)dx, where A(x) is the area of the cross sections.

The Attempt at a Solution

I have no idea how to find the area for these cross sections. I have drawn the graph and understand where the region lies. Also, they have this same exact example for square cross sections and I see how that works, but I do not know how to obtain the equation for the areas of these cross sections. Once I figure that out I know how to do the question.

On first attempt for (a) I got from x = 0 to x = 1 the area = 2h(+/-sqrt(x)) and from x = 1 to x = 3 the area = h(+/-sqrt(3 - x)), but these seem all wrong and hard to find an antiderivative for when you integrate. Please help?!

 

[mighty2000]

Thank you for your question. Let's break down each part of the problem and find the area of the cross sections for each case.

a) Rectangles of height h:
To find the area of a rectangle, we simply multiply the length and width. In this case, the length of the rectangle will be the difference between the two parabolas at a given x-value, and the width will be the given height h. So, the area of the rectangle will be: A(x) = h(x2 - (3 - 2y2)). To find the volume, we can integrate this equation from x = 0 to x = 3, since these are the limits of the region. So, the volume will be V = ∫(0 to 3) h(x2 - (3 - 2y2)) dx.

b) Equilateral triangles:
To find the area of an equilateral triangle, we first need to find the length of one side. Since the triangle is equilateral, all sides will be equal. We can use the Pythagorean theorem to find this length. The base of the triangle will be the difference between the two parabolas at a given x-value, and the height of the triangle will be the given height h. So, the length of one side will be √(h2 + (x2 - (3 - 2y2))2). To find the area, we can use the formula for the area of an equilateral triangle, which is A = √3/4 * s2, where s is the length of one side. So, the area of the cross section will be A(x) = √3/4 * (h2 + (x2 - (3 - 2y2))2). To find the volume, we can integrate this equation from x = 0 to x = 3, since these are the limits of the region. So, the volume will be V = ∫(0 to 3) √3/4 * (h2 + (x2 - (3 - 2y2))2) dx.

c) Isosceles right triangles, hypotenuse on the xy-plane:
To find the area of an isosceles right triangle, we can use the formula A = 1/2 * base * height. In this case, the base of the triangle will