Now, look at a general function f(x), as shown in (figure 4) .
suppose you want to find the area under the curve, between x=a and x=b —that is, the blue shaded area. using our method above, we can add up the areas of lots of skinny rectangles like the green one. each rectangle has area f(x)δx, and we want to sum these areas between x=a and x=b, letting δx get very small. mathematically, you may recognize this sum as a definite integral: ∫baf(x)dx.
the notation may look intimidating, but just keep in mind that the f(x)dx refers to the small rectangle areas, the ∫ sign stands, in a sense, for the s in “sum” (that’s actually how the symbol for integration was chosen), and the a and b at the bottom and top of ∫bagive you the start and end of the region you’re interested in. thus, the definite integral ∫baf(x)dx means the total area under the curve f(x) between x=a and x=b. for the function f(x) shown below, find the definite integral ∫41f(x)dx.
the figure shows the graph of the function f(x). x is measured from 0 to 5 on the horizontal axis. f(x) is measured from 0 to 6 on the vertical axis. at x equal to 0.5 the function jumps from 0 to 5, then it stays at this value till x equal to 4.5, and then the function drops to 0.

carbon is found in all living things. carbon atoms move constantly through

living organisms, the oceans, the atmosphere, and the earth’s crust in what is

known as the carbon cycle. the directions taken by carbon atoms through this

cycle are very complicated and can take millions of years to make a full circle.

all animals, from humans to the dinosaurs are part of the carbon cycle. when

animals eat food, they get carbon in the form of carbohydrates and proteins.

in animals, oxygen combines with food in the cells to produce energy for daily

activity and then gives off carbon. the carbon combines with oxygen to form

carbon dioxide (co2) and is released back into the atmosphere as a waste

product when animals breathe and exhale.

from 145-65 million years ago, earth was much

hotter than today and covered with dense,

tropical swamp forests. the trees and other

plants were immense and

provided an endless supply of

food for the giant animals that

roamed the land. somewhere in

the air above one of these

forests, a lone carbon atom has joined

up with two oxygen atoms to form a molecule of co2.

the co2 molecule was sucked into the tiny holes (stomata) on the leaf of a

fern plant and joined with sunlight, chlorophyll and water to make food and

energy in the plant’s cells through photosynthesis. the oxygen (o2) from the

co2 molecule was sent back into the atmosphere; the carbon atom (c) was

detached and used to make a molecule of sugar.

if that carbon atom had been eaten instead, a totally

different situation would have developed. let’s say

that a giant plant-eating dinosaur (herbivore) ate the

fern for breakfast, and swallowed the carbon in the

fern in the form of carbohydrates and proteins. in

the process of respiration, oxygen combined with

the carbohydrates and proteins in the dinosaur’s cells

to provide energy for its daily activity.  

12000j is work done