General slicing method Use the general slicing method to find the volume of the following solids. 7. The solid whose base is the region bounded by the curves
and
and whose cross sections through the solid perpendicular to the x-axis are squares
The Correct Answer and Explanation is:
Correct Answer: The volume of the solid is 64/15 cubic units.
Explanation
This problem requires finding the volume of a solid using the general slicing method. The process involves integrating the area of a cross-section across the length of the solid’s base.
- Determine the Base and Integration Limits:
The base of the solid is the region bounded by the curves y = x² (a parabola opening upwards) and y = 2 – x² (a parabola opening downwards). To find the extent of this region along the x-axis, the intersection points of the curves must be found. This is done by setting the two equations equal to each other:
x² = 2 – x²
2x² = 2
x² = 1
x = ±1
These intersection points, x = -1 and x = 1, will be the limits of integration. - Define the Cross-Sectional Area:
The problem specifies that the cross-sections are perpendicular to the x-axis and are squares. The side length, s, of each square at a specific x-value is the vertical distance between the two bounding curves. Within the interval [-1, 1], the curve y = 2 – x² is the upper boundary and y = x² is the lower boundary.
The side length s(x) is therefore:
s(x) = (top curve) – (bottom curve) = (2 – x²) – (x²) = 2 – 2x²
The area of a square is side², so the area of a single cross-sectional slice, A(x), is:
A(x) = [s(x)]² = (2 – 2x²)² = 4 – 8x² + 4x⁴ - Set Up and Evaluate the Volume Integral:
The volume V of the solid is the integral of the cross-sectional area A(x) from the lower limit x = -1 to the upper limit x = 1.
V = ∫[from -1 to 1] A(x) dx = ∫[from -1 to 1] (4 – 8x² + 4x⁴) dx
To evaluate this integral, find the antiderivative of the function:
∫(4 – 8x² + 4x⁴) dx = 4x – (8/3)x³ + (4/5)x⁵
Now, apply the Fundamental Theorem of Calculus using the limits of integration:
V = [4x – (8/3)x³ + (4/5)x⁵] from -1 to 1
V = [4(1) – (8/3)(1)³ + (4/5)(1)⁵] – [4(-1) – (8/3)(-1)³ + (4/5)(-1)⁵]
V = [4 – 8/3 + 4/5] – [-4 + 8/3 – 4/5]
V = 8 – 16/3 + 8/5
Finding a common denominator (15):
V = 120/15 – 80/15 + 24/15 = (120 – 80 + 24) / 15 = 64/15
