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 described in the problem, we need to integrate the areas of the cross sections along the x-axis.
First, let's find the x-values of the points where the curves intersect. Set the two equations equal to each other and solve for x:
-2x^2 + 2 = -x^2 + 1
Rearranging the equation, we get:
x^2 = 1
Taking the square root of both sides, we find:
x = ±1
So, the two curves intersect at x = -1 and x = 1.
Next, we need to find the length of the lower base of each trapezoid. In this case, the lower base is the difference in the y-values of the two curves at each x-value.
For the curve y = -2x^2 + 2, the y-value can be expressed as f(x) = -2x^2 + 2.
For the curve y = -x^2 + 1, the y-value can be expressed as g(x) = -x^2 + 1.
So, the length of the lower base can be calculated as the difference between f(x) and g(x): f(x) - g(x).
The length of the lower base = (-2x^2 + 2) - (-x^2 + 1)
= -2x^2 + x^2 + 2 - 1
= -x^2 + 1
Now let's find the upper base, which is half the length of the lower base:
Upper base = (1/2) * length of lower base
= (1/2)(-x^2 + 1)
= -1/2x^2 + 1/2
Finally, let's find the height of each trapezoid, which is twice the length of the lower base:
Height = 2 * length of lower base
= 2 * (-x^2 + 1)
= -2x^2 + 2
The area of a trapezoid can be calculated using the formula: (1/2)(base1 + base2) * height. In this case, the bases are the lower base (L) and the upper base (U).
The area of each trapezoid = (1/2)(L + U) * height
= (1/2)((-x^2 + 1) + (-1/2x^2 + 1/2)) * (-2x^2 + 2)
= (1/2)(-3/2x^2 + 3/2) * (-2x^2 + 2)
= (-3/4x^2 + 3/4) * (-2x^2 + 2)
= 6x^4 - 6x^2 + 3x^2 - 3
= 6x^4 - 3x^2 - 3
Now, we can integrate this area from x = -1 to x = 1 to find the volume of the solid:
Volume = ∫[from x = -1 to 1] (6x^4 - 3x^2 - 3) dx
Evaluating this integral will give us the volume of the solid.
I hope this helps! Let me know if you have any further questions.