# Area, Volume, and Arc Length

### Example 1

Write an integral expression for the length of the curve described by the parametric equations *x*(*t*) = *e ^{t}* and

*y*(

*t*) = 2

*t*+ 1 for 0 ≤

*t*≤ 4.

### Example 2

The equations *x*(*t*) = cos *t* and *y*(*t*) = 2sin *t* describe a parametric curve.

(a) Write an integral expression for the length of the portion of the curve graphed below.

(b) Use a calculator to evaluate your integral and explain why your answer is reasonable.

### Example 3

Write an integral expression for the parametric function described by the equations

*x*(*t*) = *x*^{3}

*y*(*t*) = *e ^{t}*sin

*t*

for 0 ≤ *t* ≤ π.

### Example 4

Write an integral expression for the length of the parametric curve

*x* = *t* cos* t*

*y* = *t* sin *t*

for α ≤ *t* ≤ β. Use the equality sin^{2} *t* + cos^{2} *t* = 1 to simplify your expression. (It will still be an integral.)