5.1 Areas Between Curves

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5.1 Areas Between Curves

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Areas Between Curves

Consider the region S that lies between two curves

y = f(x) and y = g(x) and between the vertical lines x = a

and x = b, where f and g are continuous functions and f(x) ≥ g(x) for all x in [a, b]. (See Figure 1.)

Figure 1

S = {(x, y)|a ≤ x ≤ b, g(x) ≤ y ≤ ƒ(x)}

Areas Between Curves

We divide S into n strips of equal width and then we

approximate the ith strip by a rectangle with base ∆x and height f(x_i∗) – g(x_i∗). (See Figure 2. If we like, we could take all of the sample points to be right endpoints, in which

case x_i∗ = x_i.)

(a) Typical rectangle (b) Approximating rectangles

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Areas Between Curves

The Riemann sum

is therefore an approximation to what we intuitively think of as the area of S.

This approximation appears to become better and better as n → . Therefore we define the area A of the region S as the limiting value of the sum of the areas of these

approximating rectangles.

Areas Between Curves

We recognize the limit in (1) as the definite integral of f – g.

Therefore we have the following formula for area.

Notice that in the special case where g(x) = 0, S is the region under the graph of f and our general definition of area (1) reduces.

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Areas Between Curves

In the case where both f and g are positive, you can see from Figure 3 why (2) is true:

A = [area under y = f(x)] – [area under y = g(x)]

Figure 3

Example 1

Find the area of the region bounded above by y = x² +1,

bounded below by y = x, and bounded on the sides by x = 0 and x = 1.

Solution:

The region is shown in Figure 4.

The upper boundary curve is

y = x² +1 and the lower boundary curve is y = x.

Figure 4

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Example 1 – Solution

So we use the area formula (2) with f(x) = x² + 1, g(x) = x, a = 0, and b = 1:

cont’d

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Areas Between Curves

In Figure 4 we drew a typical approximating rectangle with width ∆x as a reminder of the procedure by which the area is defined in (1).

In general, when we set up an integral for an area, it’s helpful to sketch the region to identify

the top curve y_T, the bottom curve y_B, and a typical approximating rectangle as in Figure 5.

Figure 4

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Areas Between Curves

Then the area of a typical rectangle is (y_T – y_B) ∆x and the equation

summarizes the procedure of adding (in a limiting sense) the areas of all the typical rectangles.

Notice that in Figure 5 the left-hand boundary reduces to a point, whereas in Figure 3 the right-hand boundary

reduces to a point.

Figure 3

Areas Between Curves

If we are asked to find the area between the curves y = f(x) and y = g(x) where f(x) ≥ g(x) for some values of x but g(x) ≥ f(x) for other values of x, then we split

the given region S into several regions S₁, S₂ , . . . with areas A₁, A₂ , . . . as shown in Figure 11. We then define the area of the region S to be the sum of the areas of the smaller regions S₁, S₂ , . . . , that is, A = A₁ + A₂ + . . ._. Since

f(x) – g(x) when f(x) ≥ g(x)

| f(x) – g(x)| =

g(x) – f(x) when g(x) ≥ f(x)

Figure 11

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Areas Between Curves

We have the following expression for A.

When evaluating the integral in (3), however, we must still split it into integrals corresponding to A₁, A₂,….

Example 6

Find the area of the region bounded by the curves y = sin x, y = cos x, x = 0, and x = π/2.

Solution:

The points of intersection occur when sin x = cos x, that is, when x = π/4 (since 0 ≤ x ≤ π/2). The region is sketched in Figure 12. Observe that cos x ≥ sin x when 0 ≤ x ≤ π/4 but sin x ≥ cos x when π/4 ≤ x ≤ π/2.

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Example 6 – Solution

Therefore the required area is

cont’d

Example 6 – Solution

In this particular example we could have saved some work by noticing that the region is symmetric about x = π/4 and so

cont’d

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Areas Between Curves

Some regions are best treated by regarding x as a function of y. If a region is bounded by curves with equations x = f(y), x = g(y), y = c, and y = d, where f and g are

continuous and f(y) ≥ g(y) for c ≤ y ≤ d (see Figure 13), then its area is

Figure 13

Areas Between Curves

If we write x_R for the right boundary and x_L for the left boundary, then, as Figure 14 illustrates, we have

Figure 14