Calculus AB and Calculus BC
CHAPTER 6 Definite Integrals
C. INTEGRALS INVOLVING PARAMETRICALLY DEFINED FUNCTIONS
The techniques are illustrated in Examples 22 and 23.
Evaluate where x = 2 sin θ and y = 2 cos θ.
SOLUTION: Note that dx = 2 cos θ dθ, that when x = −2, and that when x = 2.
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.
Express xy dx in terms of t if x = ln t and y = t3.
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.