MATH 123 Chapter Notes - Chapter 15: Antiderivative, Power Rule, Numerical Integration

• School: McGill University
• Department: Mathematics and Statistics
• Course: MATH 123
• Professor: Sidney Trudeau

Document Summary

The first main idea of integration is indefinite integration, or finding an antiderivative. Substitution: the quantity under a root or raised to a power; The substitution method involves choosing to be one of the following: the exponent on (cid:1857), the quantity in the denominator; The second main idea of integration is definite integration, which can be used to find the area under a curve. (cid:1858)(cid:4666)(cid:4667)(cid:1856)=lim (cid:1858)(cid:4666)(cid:4667) , where =(cid:4666)(cid:1854) (cid:1853)(cid:4667)/ and is any value of in the th interval. If (cid:1858)(cid:4666)(cid:4667) gives the rate of change of (cid:4666)(cid:4667) for in [(cid:1853),(cid:1854)], then this represents the total change in (cid:4666)(cid:4667) as goes from (cid:1853) to (cid:1854): (cid:1858)(cid:4666)(cid:4667)(cid:1856)=(cid:882, (cid:1858)(cid:4666)(cid:4667)(cid:1856)= (cid:1858)(cid:4666)(cid:4667)(cid:1856, [(cid:1858)(cid:4666)(cid:4667) (cid:1859)(cid:4666)(cid:4667)](cid:1856)= (cid:1858)(cid:4666)(cid:4667)(cid:1856) (cid:1859)(cid:4666)(cid:4667)(cid:1856, (cid:1858)(cid:4666)(cid:4667)(cid:1856)= (cid:1858)(cid:4666)(cid:4667)(cid:1856)+ (cid:1858)(cid:4666)(cid:4667)(cid:1856, (cid:1858)(cid:4666)(cid:4667)(cid:1856)= (cid:1858)(cid:4666)(cid:4667)(cid:1856) The fundamental theorem of calculus unites the two ideas of antiderivatives by showing that the way to find the area under a curve is to use the antiderivative. (cid:1858)(cid:4666)(cid:4667)(cid:1856)=(cid:4666)(cid:4667)=(cid:4666)(cid:1854)(cid:4667) (cid:4666)(cid:1853)(cid:4667), where (cid:1858) is continuous on [(cid:1853),(cid:1854)] and is any antiderivative of (cid:1858).