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Area of a Region Between 2 Curves

- Section 6.1

General Solution

- When determining the area between a function and

the x-axis - Graph the function first
- Note the zeros of the function
- Split the function into portions where f(x) gt 0

and f(x) lt 0 - Where f(x) lt 0, take absolute value of the

definite integral

Try This!

- Find the area between the function h(x)x2 x

6 and the x-axis - Note that we are not given the limits of

integration - We must determine zeros to find limits
- Also must take absolutevalue of the integral

sincespecified interval has f(x) lt 0

Area Between Two Curves

- Consider the region betweenf(x) x2 4 and

g(x) 8 2x2 - Must graph to determine limits
- Now consider function insideintegral
- Height of a slice is g(x) f(x)
- So the integral is

g

f

Area Between Curves

Find the area of the shaded region

Area Between Curves

Find the area of the shaded region

Area Between Curves

In general

Area Between Curves

In general

1. Find the area of the shaded region.

1. Find the area of the shaded region.

Find intersection points first.

2. Sketch the region represented by

2. Sketch the region represented by

top

bottom

- Representative rectangles are sometimes used.

A vertical rectangle (of width ) implies

integration with respect to x, whereas a

horizontal rectangle (of width ) implies

integration with respect to y.

- If 2 curves intersect at more that 2 points,

then to find the area of the region between the

curves, you must find all points of intersection

and check to see which curve is above the other

in each interval determined by these point.

In general, to determine the area between 2

curves, you can use

in variable x -vertical rectangles

in variable y - horizontal rectangles

Area Between Two Curves

- Sketch
- Determine which curve is on top
- Determine a and b
- Integrate!

Example

Find the area trapped between y x 2 and y 1.

y 1 y x 2

Top curve Bottom curve

Solve for a and b

x2 1, so x 1 and x 1.

The correct integral then is

Evaluate

Example

Set up an integral to find the area trapped

between y x and in the region for which x

is between 2 and 5.

Can we do this exactly?

Example 6 Determine the area of the region

enclosed by

and .

The area is,

1. Find the area of the region bounded by

No calculator.

1. Find the area of the region bounded by

No calculator.

2. Use your calculator to find the area between

2. Use your calculator to find the area between

A

Intersection points

C

B

Store these values as the respective letter!

1. Find the area of the region bounded by

No calculator.

1. Find the area of the region bounded by

No calculator.

2. Find the area of the triangle with vertices

A(2,-3), B(4,6), and C(6,1).

2. Find the area of the triangle with vertices

A(2,-3), B(4,6), and C(6,1).

Integration as an Accumulation Process

- We can think of this as a function of b
- This gives us the accumulated area under the

curve on the interval 0, b

Try It Out

- Find the accumulation function for
- Evaluate
- F(0) 0
- F(4) 3
- F(6) 16

AP QUESTIONS

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