# Symmetrical Areas

In the realm of geometry and spatial analysis, the concept of symmetrical areas holds a fundamental significance. Symmetry, as a mathematical principle, governs the balance and proportionality inherent in shapes and structures. When applied to areas, symmetry becomes a powerful tool for understanding the equilibrium between different regions within a given space. In this exploration, we delve into the intricate world of symmetrical areas, unraveling their mathematical underpinnings and exploring their diverse applications across various disciplines. This examination not only sheds light on the inherent beauty of symmetry but also underscores its practical implications in design, architecture, and beyond.

- What are the symmetry properties of finding areas using integrals?
- What is the smallest parameter possible for a rectangle whose area is 16 square inches and what are it’s dimensions?
- Find the dimensions of the rectangle of maximum area whose perimeter is 16 cm ?
- How do you find the area under the graph of #f(x)=x^3# on the interval #[-1,1]# ?
- An open box has a square base and a surface area of 240 square inches. What dimensions (width × length × height) will produce a box with maximum volume?
- Find the rectangle with the maximum area, which can be turned in the corner. ?
- How do you find the area of the region bounded by the graph of #f(x)=1-x^2# and the #x#-axis on the interval #[-1,1]# ?
- How do you find the area under the graph of #f(x)=sin(x)# on the interval #[-pi,pi]# ?
- A box is to build with a square base and an open top. Material for the base costs #$4/m^2#, while the material for the sides costs #$2/m^2#. Find the dimensions of the box of maximum volume which can be built at a cost of #$1200#?
- A solid rectangular block has a square base .find its maximum volume if the sum of the height and any one side of the base is 12cm ?
- Find the value of the surface integral (2x^2yi - y^2j + 4xz^2k).dS where is S the surface of the cylinder y^2 + z^2 = 9 bounded by the planes x=0 and x=2?
- How do you find the area of the region bounded by the graph of #f(x)=x^4# and the #x#-axis on the interval #[-2,2]# ?
- How do you Evaluate the integral by changing to cylindrical coordinates?
- How do you find the area under the graph of #f(x)=cos(x)# on the interval #[-pi/2,pi/2]# ?
- A rectangular piece canvass with dimensions 10m by 6m is used to make a pool.Equal sizes squares are to be cut from each corner and remaining will folded up around some plastic tubing.what is the dimension of the pool so the water volume is maximum?
- How do you find the area under the graph of #f(x)=x^2# on the interval #[-3,3]# ?
- A rectangular box with a square base and no top is to have a volume of 108 cubic cm. Find the dimensions for the box that will require the least number of material?
- An open box is to be made from a rectangular sheet of cardboard od dimension 16cm by 24cm by cutting out squares of = size from each of the four corners & bending up flaps.Find the dimensions of box of largest volume that can be made?
- How do you solve the following questions?
- How to solve this very difficult volume problem involving integration?