Cross Sections and Volumes

Cross sections are the intersections of a solid with a plane, which can be used to analyze the shape and volume of the solid.

The area A of an equilateral triangle is given by the formula: $$A = \frac{\sqrt{3}}{4}s^2$$ .

To find the volume V, integrate the area A of the cross-section with respect to the axis of integration: $$V = \int A(x) \, dx$$ .

The volume V can be calculated using the formula: $$V = \int_{a}^{b} \frac{\pi}{4} (f(x))^2 \, dx$$ .

The base is the region in the plane that is rotated around an axis to create the solid.

The area A of a semicircle is given by the formula: $$A = \frac{\pi}{2} r^2$$ .

The volume V is found by integrating the area of the square cross section: $$V = \int_{a}^{b} (s(x))^2 \, dx$$ , where s(x) is the side length of the square.

For semicircular cross sections, the radius is equal to half the height of the cross section at that point.

The limits of integration are found by solving the equations of the curves to find their points of intersection.

The volume V can be calculated using the formula: $$V = \int_{a}^{b} \frac{1}{2} b(x) h(x) \, dx$$ , where b(x) is the base and h(x) is the height of the triangle.