Consider a planar region bounded by two curves that are both written as functions of the vertical variable, y. The left and right boundary curves are continuous between y = c and y = d, and these two horizontal lines define the vertical limits of the region. Because the boundaries depend on y rather than x, the area is most appropriately evaluated using horizontal slices.The area is obtained using the Riemann sum method. The region is divided into many thin horizontal strips, each having an...

Consider two continuous functions defined on a closed interval from a to b. The region between these curves is bounded vertically by their graphs and horizontally by the endpoints of the interval. The objective is to measure the area of this region.An initial estimate of the area can be obtained by dividing the interval into a large number of narrow vertical strips of equal width. Each strip is approximated by a rectangle whose height is given by the vertical difference between the two...

Spontaneous Chemical Reactions
Spontaneous redox reactions occur abundantly in nature. The chemical reaction occurring in a disposable AA battery powering our remote controls is one such example of a spontaneous redox reaction. Another example is the immersion of coiled copper wire into an aqueous silver nitrate solution. The reaction shows a gradual, visually impressive color change from colorless to bright blue and the formation of a grey precipitate on the copper wire. In this experiment,...

Social traps are negative situations where people get caught in a direction or relationship that later proves to be unpleasant, with no easy way to back out of or avoid. The concept was orignally introduced by John Platt who applied psychology to Garrett Hardin's "Tragedy of the Commons", where in New England herd owners could let their cattle graze in the common ground. This situation seems like a good idea, but an individual could have an advantage. If they owned...

Allosteric proteins have more than one ligand binding site; the binding of a ligand to any of these sites influences the binding of ligands to the other sites. When a protein is allosteric, its binding sites are called coupled or linked.  In the case of enzymes, the site that binds to the substrate is known as the active site and the other site is known as the regulatory site. When a ligand binds to the regulatory site, this leads to conformational changes in the protein that can influence...

Cooperative allosteric transitions can occur in multimeric proteins, where each subunit of the protein has its own ligand-binding site. When a ligand binds to any of these subunits, it triggers a conformational change that affects the binding sites in the other subunits; this can change the affinity of the other sites for their respective ligands. The ability of the protein to change the shape of its binding site is attributed to the presence of a mix of flexible and stable segments in the...