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Area Between Curves

Lesson #2 of

Unit 6. Applications of Integration

(Textbook 6.1)

Area Between Curves (1/3)

Previously, we have considered an area under a curve using Riemann Sum, and obtained the result…

Now we consider a region between curves.

Each rectangle’s height of Riemann Sum for an area under a curve was the value of .

Then for an area between two curves, the height is the difference between two curves, which is as shown on the left.

Text p.422

Area Between Curves (2/3)

Applying the difference of two functions, , as the height of each rectangle, we obtain following result for the calculation of area between two curves.

Above is the definite integral of f – g, so we generalize it as following.

Text p.422

Example

Find the area of the region bounded above by y = e x, bounded below by y = x, and bounded on the sides by x = 0 and x = 1.

Sketch the region enclosed by the parabolas and , and then find the area.

Text p.423

Practice

Sketch the region enclosed by the curves, and then find the area.

Text p.600

Area Between Curves (3/3)

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, then its area is…

Text p.426

Find the area of the region enclosed by the parabolas and

Practice

Sketch the region enclosed by the curves, and then find the area.

Text p.427

Practice

Use calculus to find the area of the triangle with the given vertices.

(0, 0), (3, 1), (1, 2)

Text p.427

Area Between Curves: The Split Case

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 x of but g (x) f (x) for other values of x, then we split the given region S into several regions S1, S2, …

We then define the area of the region S to be the sum of the areas of the smaller regions S1, S2, ... Since…

f (x) – g (x) when f (x) g (x)

| f (x) – g (x) | =

g (x) – f (x) when g (x) f (x)

the area between the curves y = f (x) and y = g (x) and between x = a and x = b is…

Text p.426

Example

Find the area of the region bounded by the curves , , , and .

Text p.423

Practice

Sketch the region enclosed by the curves, and then find the area.

Text p.600

Suggested Problems

Textbook p.427

Sketch the region and find the area: 1, 5, 8, 16

Sketch the region and find the area: 2, 12, 20, 30

Sketch the region and find the area: 21, 22, 27

Suggested Problems - SOLUTIONS -

Textbook p.427

1) 5)

8) 16)

2) 12)

20) 30)

Suggested Problems - SOLUTIONS -

Textbook p.427

21) OR

by symmetry

22) By symmetry,

27)