When using parametric equations we must be sure to express everything in terms of the parameter.
In Example 22 we replaced in terms of θ: (1) the integrand, (2) dx, and (3) both limits. Remember
that we have defined dx as x ′(θ) dθ.
EXAMPLE 23
Express xy dx in terms of t if x = ln t and y = t^3.
SOLUTION:
We see that We now find limits of integration in terms of t:
For x = 0, we solve ln t = 0 to get t = 1.
For x = 1, we solve ln t = 1 to get t = e.D. DEFINITION OF DEFINITE INTEGRAL AS THE LIMIT OF
A SUM: THE FUNDAMENTAL THEOREM AGAIN
Most applications of integration are based on the FTC. This theorem provides the tool for evaluating
an infinite sum by means of a definite integral. Suppose that a function f (x) is continuous on the
closed interval [a, b]. Divide the interval into n subintervals of lengths* Δxk. Choose numbers, one in
each subinterval, as follows: x 1 in the first, x 2 in the second, ..., xk in the kth, ..., xn in the nth. Then
Any sum of the form is called a Riemann sum.
AREA
If f (x) is nonnegative on [a, b], we see (Figure N6–1) that f (xk) Δxk can be regarded as the area of a
typical approximating rectangle. As the number of rectangles increases, or, equivalently, as the width
Δx of the rectangles approaches zero, the rectangles become an increasingly better fit to the curve.
The sum of their areas gets closer and closer to the exact area under the curve. Finally, the area
bounded by the x-axis, the curve, and the vertical lines x = a and x = b is given exactly by