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# The Area Between Two Curves Help

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By McGraw-Hill Professional
Updated on Sep 1, 2011

## Introduction to The Area Between Two Curves

Frequently it is useful to find the area between two curves. See Fig. 4.17. Following the model that we have set up earlier, we first note that the intersected region has left endpoint at x = a and right endpoint at x = b . We partition the interval [ a , b ] as shown in Fig. 4.18. Call the partition

P = { x 0 , x 1 , ..., x k }.

Fig. 4.17

Fig. 4.18

Then, as usual, we erect rectangles over the intervals determined by the partition (Fig. 4.19).

Notice that the upper curve, over the interval [ a , b ], is y = f ( x ) and the lower curve is y = g ( x ) (Fig. 4.17). The sum of the areas of the rectangles is therefore

But of course this is a Riemann sum for the integral

Fig. 4.19

We declare this integral to be the area determined by the two curves.

#### Example 1

Find the area between the curves y = x 2 − 2 and y = −( x − 1) 2 + 3.

#### Solution 1

We set the two equations equal and solve to find that the curves intersect at x = −1 and x = 2. The situation is shown in Fig. 4.20 . Notice that y = − ( x −1)2 + 3 is the upper curve and y = x2 − 2 is the lower curve. Thus the desired area is

The area of the region determined by the two parabolas is 9.

Fig. 4.20

#### Example 2

Find the area between y = sin x and y = cos x for π/4 ≤ x ≤ 5π/4.

#### Solution 2

On the given interval, sin x ≥ cos x . See Fig. 4.21 . Thus the area we wish to compute is

Fig. 4.21

You Try It: Calculate the area between y = sin x and y = cos x , −π ≤ x ≤ 2π.

You Try It: Calculate the area between y = x2 and y = 3 x + 4.

Find practice problems and solutions for these concepts at: The Integral Practice Test.

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