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Area, Volume, and Arc Length

Area, Volume, and Arc Length

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 = x4 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 x2 + y2 = 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 = ex, 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 – x2 and below the graph x2 + y2 = 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 – x2. 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 – x2. Write an integral expression for the area of R, using horizontal slices.

Example 8

Let R be the region between the graphs , y = x2, 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 – x2. 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 – x2. 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 = x4 + 1 and y = x2 + 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.

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