Express the length of the curve
with parametric equations 𝑥 equals 𝑡 squared minus 𝑡 and 𝑦 equals 𝑡 to the
fourth power, where 𝑡 is greater than or equal to one and less than or equal to
four, as an integral.
We recall that the formula for the
arc length 𝐿 of a curve defined parametrically between the limits of 𝑡 equals 𝛼
and 𝑡 equals 𝛽 is the definite integral between 𝛼 and 𝛽 of the square root of
d𝑥 by d𝑡 squared plus d𝑦 by d𝑡 squared with respect to 𝑡. Now, our curve is defined
parametrically by 𝑥 equals 𝑡 squared minus 𝑡 and 𝑦 equals 𝑡 to the fourth
power. And we want to find this arc length
between the limits of 𝑡 equals one and 𝑡 equals four. So we let 𝛼 be equal to one, 𝛽 be
equal to four. And we see we’re going to need to
differentiate 𝑥 and 𝑦 with respect to 𝑡.
Now, to differentiate a polynomial
term, we simply multiply the entire term by the exponent and then reduce the
exponent by one. So the derivative of 𝑡 squared is
two 𝑡. And when we differentiate negative
𝑡, we get negative one. d𝑥 by d𝑡 is, therefore, two 𝑡 minus one. And this satisfies the criteria
that the derivative of this function is continuous. d𝑦 by d𝑡 is the first
derivative of 𝑡 to the fourth power. That’s four 𝑡 cubed, which is also
a continuous function. Now, we noticed that we’re going to
have to square these in our formula for the arc length. So let’s work out d𝑥 by d𝑡
squared and d𝑦 by d𝑡 squared before substituting into the formula.
By distributing the parentheses, we
find that two 𝑡 minus one all squared is four 𝑡 squared minus four 𝑡 plus
one. And four 𝑡 cubed all squared is
16𝑡 to the sixth power. Our final step is to substitute
into the formula for the arc length. And we find that 𝐿 is equal to the
definite integral between one and four of the square root of four 𝑡 squared minus
four 𝑡 plus one plus 16𝑡 to the sixth power d𝑡. We might choose to rewrite the
expression inside our root in descending powers of 𝑡. And when we do, we find that the
arc length of the curve defined by our parametric equations for 𝑡 is greater than
or equal to one and less than or equal to four is the integral shown.