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.
The base is enclosed by the two parabolas, each with roots at (-1,0) and (1,0)
The lower base B lies between the curves, with length 2(1-x^2)-(1-x^2) = 1-x^2
Using symmetry, then the volume is
2∫[0,1] (B+b)/2 * h dx
= 2∫[0,1] ((1-x^2) + 1/2 (1-x^2))/2 * 2(1-x^2) dx
= 2∫[0,1] 3/2 (1-x^2)^2 dx
= 3∫[0,1] (1-x^2)^2 dx
= 8/5
thanks
Ah, a solid filled with trapezoids, huh? Sounds like a geometry-themed circus act! Let's get cracking on finding the volume.
To start, we'll need to find the x-coordinates of the points where the curves intersect. So, we'll set -2x^2 + 2 equal to -x^2 + 1:
-2x^2 + 2 = -x^2 + 1
Rearranging a bit:
x^2 - 1 = 0
Now, we can solve for x by factoring:
(x + 1)(x - 1) = 0
So, we have x = -1 and x = 1 as the x-coordinates of the points of intersection.
Next, we'll integrate the area of each of our trapezoids from x = -1 to x = 1.
The lower base length is given by the difference between the curves at each x-value. So, for any x within our limits:
Lower base length = (-2x^2 + 2) - (-x^2 + 1)
Simplifying, we get:
Lower base length = -x^2 + 1
Now, the upper base length is half of the lower base length:
Upper base length = (1/2) * (-x^2 + 1)
And lastly, the height is twice the length of the lower base:
Height = 2 * (-x^2 + 1)
To find the area of each trapezoid, we can use the formula:
Area = (lower base + upper base) * height / 2
Plugging in our values, we get:
Area = (-x^2 + 1 + (1/2) * (-x^2 + 1)) * (2 * (-x^2 + 1)) / 2
Simplifying further:
Area = (-3/2) * x^4 + 3 * x^2 - 1
Now, we integrate this area from x = -1 to x = 1 to find the total volume:
Volume = ∫[from -1 to 1] ((-3/2) * x^4 + 3 * x^2 - 1) dx
Evaluating this integral will give us the volume of the solid shaped like trapezoids. So, let's keep our math circus act going until we get the answer!
To find the volume of the solid, we need to integrate the areas of the cross-sections along the x-axis.
Let's break down the problem into smaller steps:
1. Find the intersection points of the two curves to determine the limits of integration.
To find the intersection points, we set the equations equal to each other:
-2x^2 + 2 = -x^2 + 1
Simplifying the equation, we get:
x^2 = 1
Taking the square root of both sides, we find:
x = 1 and x = -1
So the limits of integration will be -1 to 1.
2. Find the length of the lower base (b1) of the trapezoid at each x-value.
The lower base of the trapezoid is given by the difference between the y-coordinates of the two curves at a given x-value.
b1 = (-2x^2 + 2) - (-x^2 + 1)
Simplifying further:
b1 = -2x^2 + 2 + x^2 - 1
= -x^2 + 1
3. Find the length of the upper base (b2) of the trapezoid at each x-value.
The upper base of the trapezoid is equal to half the length of the lower base.
b2 = (1/2) * b1
= (1/2) * (-x^2 + 1)
4. Find the height (h) of the trapezoid at each x-value.
The height of the trapezoid is equal to two times the length of the lower base.
h = 2 * b1
= 2 * (-x^2 + 1)
5. Calculate the area of each trapezoid cross-section.
The area of a trapezoid is given by the formula: A = (1/2) * (b1 + b2) * h
= (1/2) * (-x^2 + 1 + (1/2) * (-x^2 + 1)) * 2 * (-x^2 + 1)
= (-x^2 + 1 + (-1/2)x^2 + 1/2) * 2 * (-x^2 + 1)
= (-3/2)x^4 + 2x^2 - 3/2
6. Integrate the area function to find the volume.
The volume is given by the integral of the area function with respect to x, from -1 to 1:
V = ∫[from -1 to 1] (-3/2)x^4 + 2x^2 - 3/2 dx
Evaluating this integral will give you the volume of the solid.
Note: The integration can be done using techniques such as the Power Rule and the Constant Multiple Rule.