As you can see, the technique is very simple. First, you find the side of the cross-section in terms of y.
This will involve a vertical slice. Then, you plug the side into the equation for the area of the cross-
section. Then, integrate the area from one endpoint of the base to the other. On the AP Exam, cross-
sections will be squares, equilateral triangles, circles, or semi-circles, or maybe isosceles right triangles.
So here are some handy formulas to know.
Given the side of an equilateral triangle, the area is A = (side)^2.Given the diameter of a semi-circle, the area is A = (diameter)^2.Given the hypotenuse of an isosceles right triangle, the area is A = .Example 9: Use the same base as Example 8, except this time the cross-sections are equilateral triangles.
We find the side of the triangle just as we did above. It is 2y, which is . Now, because the area
of an equilateral triangle is (side)^2 , we can find the volume by evaluating the integral (4)(4 −
x^2 ) dx = (4 − x^2 ) dx.
We get (4 − x^2 ) dx = .
Example 10: Use the same base as Example 8, except this time the cross-sections are semi-circles whose
diameters lie on the base. We find the side of the semi-circle just as we did above. It is 2y, which is