PRACTICE PROBLEMS:
1. Find, by integration, the area under the curve
y=x+4
bounded by the X axis and the lines
x=2
and
x=7
verify this by a geometric process.
2. Find the area under the curve
bounded by the X axis and the lines
x = 0
and
x = 2
3. Find the area between the curve
and the X axis, from
x = 1
to
x = 3
ANSWERS:
SUMMARY
The following are the major topics covered in this
chapter:
1. Definition of integration: Integration is defined as the inverse of differentiation.
where F(x) is the function whose derivative is the
function f(x);
is the integral sign; f(x) is the integrand; and dx is the differential.
2. Area under a curve:
where
(sigma) is the symbol for sum, n is the number of rectangles,
is the area of each rectangle, and k is the
designation number of each rectangle.
Intermediate
Value Theorem:
where f(c) is the function at an
intermediate point between a and b.
where F(b)  F(a) are the integrals of the
function of the curve at the values b and a.
3. Indefinite integrals:
where C is called a constant of integration, a number which is independent of the variable of integration.
Theorem 1. If two functions differ by a constant, they have the
same derivative.
Theorem 2. If two functions have the same derivative, their
difference is a constant.
4. Rules for integration:
The integral of a differential of a function is the
function plus a constant.
A constant may be moved across the integral sign. NOTE: A
variable may NOT be moved across the integral sign.
The integral of
du may be obtained by adding 1 to the exponent
and then dividing by this new exponent. NOTE: If n is minus
1, this rule is not valid.
The integral of a sum is equal to the sum of the
integrals.
5. Definite integrals:
where b, the upper limit, and a, the
lower limit, are given values.
6. Areas above and below a curve:
If the graph of y = f(x), between x = a and x =
b, has portions above and portions below the X axis, then
is the sum of the absolute values of the
positive areas above the X axis and the negative areas below the X axis.
ADDITIONAL PRACTICE PROBLEMS Evaluate the following
integrals:
ANSWERS TO ADDITIONAL PRACTICE PROBLEMS
