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Area, Volume, and Arc Length

Area, Volume, and Arc Length

At a Glance - Assumptions When Finding Area

We're going to be doing a lot of problems making fish sticks or pizza slices or linguine pasta out of different foods. Since mathematicians are lazy, and because we should be focused more on our knife skills–we like our fingers–we're going to make some assumptions to save time and space later.

From now on, assume that

  • all vertical slices have width Δ x and all horizontal slices have thickness Δ y. That way we don't have to keep saying "take a vertical slice of a guava with width Δ x."
      
  • when we say "the height of the slice is ..." we mean "the approximate height of the slice is ...". Similarly when we say "the area of the slice is ..." we mean "the approximate area of the slice is ..." That way we don't have to keep saying the word "approximately." As much as we like that word, we're going to sound like a Star Wars drone if we keep saying it.
      
  • once we know the area of a slice, we can write down the integral that gives the area of the whole region. We don't have to write "we add up all the approximate regions and take the limit as the number of slices goes to ∞ to get an integral" every single time.
      
  • if a problem asks us to "write an integral expression" for the area of a region, that means we don't need to evaluate the integral. If you want to practice evaluating integrals, then by all means, feel free to do them. We're focusing on how to set up the integrals in this chapter, which is the most difficult part for most people. We can save ourselves the headache of solving them for another day.

Example 1

Let R be the region bounded by the graph of y = x2 and the line y = 4. Write an integral expression for the area of R, using

(a) vertical slices

(b) horizontal slices


Example 2

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 horizontal slices.


Exercise 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


Exercise 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


Exercise 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


Exercise 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


Exercise 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.


Exercise 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.


Exercise 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.


Exercise 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.


Exercise 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.


Exercise 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.


Exercise 11

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


Exercise 12

Let R be the region shown below.

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


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