Hi, I have a calculus question that I just cannot figure out, it is about volume of cross sections.
I would very much appreciate it if someone could figure out the answer and show me all the steps.
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 the lower base.
Thanks in Advance :D
To find the volume of the solid, we can use the method of cross-sectional areas.
First, let's find the equations of the curves and determine the limits of integration. We are given the curves y = -2x^2 + 2 and y = -x^2 + 1.
To find the intersection points of the curves, we can set them equal to each other: -2x^2 + 2 = -x^2 + 1.
Simplifying, we get: -x^2 + 1 = 0.
Rearranging, we have: x^2 = 1.
Taking the square root of both sides, we get: x = ±1.
So, the limits of integration will be from x = -1 to x = 1.
Now, let's find the area of a cross section at an arbitrary x-value.
The lower base of each trapezoid is the difference between the y-values of the curves y = -2x^2 + 2 and y = -x^2 + 1 at that x-value. Therefore, lower base = (-2x^2 + 2) - (-x^2 + 1) = -x^2 + 1.
The upper base is equal to 1/2 the length of the lower base, so upper base = (1/2)(-x^2 + 1) = -0.5x^2 + 0.5.
The height of each trapezoid is given as 2 times the length of the lower base, so height = 2(-x^2 + 1) = -2x^2 + 2.
Now, we can use these dimensions to calculate the area of the trapezoid using the formula: A = (upper base + lower base) * height / 2.
Substituting the values we found, we get: A = ((-0.5x^2 + 0.5) + (-x^2 + 1)) * (-2x^2 + 2) / 2.
Simplifying this expression gives: A = (-1.5x^2 + 1.5) * (-2x^2 + 2) / 2.
To find the volume, we need to integrate the cross-sectional areas from x = -1 to x = 1.
So, the volume of the solid is given by the integral: V = ∫[from -1 to 1] ((-1.5x^2 + 1.5) * (-2x^2 + 2) / 2) dx.
Evaluating this integral will give you the volume of the solid.