Applications of Integrals: Volumes and Moments Using Cross-Sections

Volumes and Moments Using Cross-Sections

Volume of a Solid with Square Cross-Sections

When a solid has a base bounded by two curves and cross-sections perpendicular to the x-axis are squares, the volume can be found by integrating the area of each square along the interval.

• Base Region: The region is bounded above by y = x and below by y = x^4 for 0 \leq x \leq 1.

• Side Length of Square: At each x, the side length is the vertical distance between the curves: s = x - x^4.

• Area of Cross-Section: The area of each square is A(x) = (x - x^4)^2.

• Volume Integral: The total volume is the integral of the area from x = 0 to x = 1:

$V = \int_0^1 (x - x^4)^2 dx$

• Expanding and integrating gives:

$V = \int_0^1 (x^2 - 2x^5 + x^8) dx = \left[ \frac{x^3}{3} - \frac{2x^6}{6} + \frac{x^9}{9} \right]_0^1 = \frac{1}{3} - \frac{1}{3} + \frac{1}{9} = \frac{1}{9}$

• Application: This method is commonly used to find the volume of solids with known cross-sectional shapes (e.g., squares, rectangles, semicircles) perpendicular to an axis.

Moment About the x-Axis for a Lamina

The moment about the x-axis (Mx) for a thin plate (lamina) of uniform density bounded by two curves can be found by integrating the product of the y-coordinate of the centroid of each thin slice and its area.

• Region: Bounded by y = x^2 and y = 3x from x = 0 to x = 3.

• Centroid of Slice: The y-coordinate of the centroid at each x is the average of the two bounding functions: \frac{1}{2}(3x + x^2).

• Area of Slice: The area is the vertical distance between the curves: 3x - x^2.

• Moment of Slice: Multiply the centroid by the area and integrate:

$M_x = \int_0^3 \frac{1}{2}(3x + x^2)(3x - x^2) dx$

• Expanding and integrating gives:

$M_x = \int_0^3 \frac{1}{2}(9x^2 - x^4) dx = \frac{1}{2} \left[ 3x^3 - \frac{1}{5}x^5 \right]_0^3 = \frac{1}{2}(81 - \frac{243}{5}) = \frac{81}{5}$

• Application: The moment about the x-axis is used to find the center of mass and analyze the distribution of mass in a planar region.

Additional info: These problems illustrate the use of definite integrals to compute geometric and physical properties of regions bounded by curves, a central topic in calculus applications.