## Calculus AB and Calculus BC

## CHAPTER 6 Definite Integrals

### C. INTEGRALS INVOLVING PARAMETRICALLY DEFINED FUNCTIONS

The techniques are illustrated in Examples 22 and 23.

**BC ONLY**

**EXAMPLE 22**

Evaluate where *x* = 2 sin θ and *y* = 2 cos θ.

**SOLUTION**: Note that *dx* = 2 cos θ *d*θ, 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) *d*θ.

**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.