1

–

3

√x3–1

interest 275

y

–

b

x

–

a







T

When computing a definite integral via this tech-

nique, it is advisable to compute the indefinite integral

first and then, at the end of the process, work with the

limits of integration. For example, the definite integral

∫5

1x2dx does not equal ∫5

1√

–

udu, since the

values x= 1 to x= 5 refer to the x-variable and not to

the new u-variable. We compute:

See also

ANTIDIFFERENTIATION

;

INTEGRAL CALCU

-

LUS

;

INTEGRATION BY PARTS

.

intercept A point at which two figures intersect is

called an intercept. The term is usually reserved for

the locations at which the

GRAPH OF A FUNCTION

crosses the x-axis and the y-axis in a system of

C

ARTESIAN COORDINATES

, to form what are called the

x- and y-intercepts. If a straight line has x-intercept

(a, 0) and y-intercept (0, b), then the equation of the

line is given by + = 1. This is called the “intercept

form” of its equation.

The y-intercept of an arbitrary function y= f(x) is

the point (0, f(0)).

interest A fee paid for the use of money is called

interest. For instance, mortgage companies charge home

buyers a fee for borrowing money, and credit card com-

panies charge customers a fee for the privilege of using

company money to make purchases. In reverse, banks

pay customers money for maintaining a balance in a

savings account. (The customers have, in effect, lent

money to the institution.) In any such arrangement, the

amount borrowed or invested is called the principal or

capital, and the fee, expressed as a percentage rate, that

is, as a number of dollars per hundred to be paid each

year of the loan, is called the interest rate.

There are two types of interest. The first, simple

interest, is computed only on the principal. For exam-

ple, if a customer borrows $3,000 for 3 years under

simple interest computed at 15 percent per annum, then

the customer has agreed to pay 0.15 ×3,000 = $450 for

each year of the loan, plus return the original $3,000 at

the end of the 3-year period. The customer thus pays a

total of $4,350 at the end of the 3 years.

In general, the total interest Ipaid on a principal

amount Pat an annual interest rate R(expressed as a

decimal) for Tyears is given by the formula:

I= P×R×T

(In our example, I= 3,000 ×0.15 ×3 = 1,350.) At the

end of the loan, the amount Aowed is:

A= P+ I= P+ P×R×T= P(1 + RT)

It is rare today, however, that a lending institution will

provide loans with fees computed by simple interest.

Typically, one is also expected to pay interest on any

interest incurred as the loan progresses.

Compound interest is calculated by adding the

interest to the principal and recalculating the interest

at the end of agreed “conversion periods.” For exam-

ple, a credit card company may charge 18 percent

interest per annum, but will compound the interest

monthly (at a rate of 18/12 = 1.5% per month).

Thus, after borrowing $1,000, say, a customer is

expected to return the principal plus 0.015×1,000 =

$15 in interest, that is, a total of $1,015, at the end

of the first month. In failing to do so, the customer

will then be expected to return this amount, plus an

additional 1.5% on this amount, that is, a total of

1,015 ×(1 + 0.015) = $1,030.23 at the end of the

second month. At the end of each month passed,

the amount owed increases by a factor of

. After 1 year, that is 12 months, the balance

on the credit card will thus be =

$1,195.62.

In general, if a principal amount of Pdollars is bor-

rowed at an interest rate of Rpercent per year

(expressed as a decimal), and the interest is compounded

ntimes a year for Tyears, then the amount Aowed at

the end of that time period is given by the formula: