# Area, Volume, and Arc Length

# Assumptions When Finding Area Exercises

### Example 1

Let *R* be the region above the line and below the graph of *y* = cos *x* on the interval

Write an integral expression for the area of *R*,

(a) vertical slices

(b) horizontal slices

### Example 2

Let *R* be the region bounded by the curve *y* = *x*^{4} and the lines *x* = 2 and *y* = 81. Write an integral expression for the area of *R*, using

(a) vertical slices

(b) horizontal slices

### Example 3

Let *R* be the region in the first quadrant bounded by the graph of *x*^{2} + *y*^{2} = 1 and the *x*- and *y*- axes. Write an integral expression for the area of *R*, using

(a) vertical slices

(b) horizontal slices

### Example 4

Let *R* be the region bounded by the graphs of *y* = *e ^{x}*,

*y*= 1, and

*x*= 3.

Write an integral expression for the area of *R*, using

(a) vertical slices

(b) horizontal slices

### Example 5

Let *R* be the region above the graph of *y* = 1 – *x*^{2} and below the graph *x*^{2} + *y*^{2} = 1. Write an integral expression for the area of *R*, using vertical slices.

### Example 6

Let *R* be the region in the first quadrant bounded by the *x*-axis and the graphs and *y* = 5 – *x*^{2}. Write an integral expression for the area of *R*, using horizontal slices.

### Example 7

Let *R* be the region in the first quadrant bounded by the line *y* = 5 and the graphs and *y* = 5 – *x*^{2}. Write an integral expression for the area of *R*, using horizontal slices.

### Example 8

Let *R* be the region between the graphs , *y* = *x*^{2}, and the line *x* = 4.

Write an integral expression for the area of *R*, using vertical slices.

### Example 9

Let *R* be the region in the first quadrant bounded by the *x*-axis and the graphs and *y* = 5 – *x*^{2}. Write an integral expression for the area of *R*, using vertical slices.

### Example 10

Let *R* be the region in the first quadrant bounded by the line *y* = 5 and the graphs and *y* = 5 – *x*^{2}. Write an integral expression for the area of *R*, using vertical slices.

### Example 11

Let *R* be the region shown below. Write an integral expression for the area of *R*, using horizontal slices.

### Example 12

Let *R* be the region between the graphs of *y* = *x*^{4} + 1 and *y* = *x*^{2} + 1 for 1 ≤ *y* ≤ 5. Write an integral expression for the area of *R*, using

(a) vertical slices

(b) horizontal slices

### Example 13

Let *R* be the region shown below.

Write an integral expression for the area of *R*, using vertical slices.