Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

### Rectangular Approximations

If *f* ≥ 0, the area under the curve of *f* can be approximated using three common types of rectangles: left-endpoint rectangles, right-endpoint rectangles, or midpoint rectangles. (See Figure 12.2-1.)

The area under the curve using n rectangles of equal length is approximately:

If *f* is increasing on [*a*, *b*], then left-endpoint rectangles are inscribed rectangles and the right-endpoint rectangles are circumscribed rectangles. If *f* is decreasing on [*a*, *b*], then left-endpoint rectangles are circumscribed rectangles and the right-endpoint rectangles are inscribed. Furthermore,

inscribed rectangle ≤ area under the curve ≤ circumscribed rectangle.

#### Example 1

Find the approximate area under the curve of *f* (*x* ) = *x*^{2} +1 from *x* =0 to *x* =2, using 4 left-endpoint rectangles of equal length. (See Figure 12.2-2.)

Let Δ*x*_{i} be the length of *i*th rectangle. The length .

Area under the curve .

Enter Σ (((0.5(*x* ≤ 1))^{2} +1) * 0.5, *x*, 1, 4) and obtain 3.75.

Or, find the area of each rectangle:

- Area of Rect

_{I}= (

*f*(0))Δ

*x*

_{1}= (1) = .

- Area of Rect

_{II}= (

*f*(0.5))Δ

*x*

_{2}= ((0.5)

^{2}+ 1) = 0.625.

- Area of Rect

_{III}= (

*f*(1))Δ

*x*

_{3}= ((1

^{2}+ 1) = 1.

- Area of Rect

_{IV}= (

*f*(1.5))Δ

*x*

_{4}=((1.5)

^{2}+ 1) = 1.625.

Area of (Rect_{I} + Rect_{II} + Rect_{III} + Rect_{IV}) = 3.75.

Thus, the approximate area under the curve of *f*(*x* ) is 3.75.

#### Example 2

Find the approximate area under the curve of *f* (*x* = from *x* = 4 to *x* = 9 using 5 rightendpoint rectangles. (See Figure 12.2-3.)

Let Δ*x*_{i} be the length of ith rectangle. The length Δ*x*_{i} = = 1; *x*_{i} = 4 + (1)*i* = 4 + *i*.

Or, using Σ notation:

Thus the area under the curve is approximately 13.160.

#### Example 3

The function *f* is continuous on [1, 9] and *f* > 0. Selected values of *f* are given below:

Using 4 midpoint rectangles, approximate the area under the curve of *f* for *x* =1 to *x* =9. (See Figure 12.2-4.)

Let Δ*x*_{i} be the length of *i*th rectangle. The length Δ*x*_{i} = 2.

- Area of Rect

_{I}=

*f*(2)(2) = (1.41)2 = 2.82.

- Area of Rect

_{II}=

*f*(4)(2) = (2)2 = 4.

- Area of Rect

_{III}=

*f*(6)(2) = (2.45)2 = 4.90.

- Area of Rect

_{IV}=

*f*(8)(2) = (2.83)2 = 5.66.

Area of (Rect_{I} + Rect_{II} + Rect_{III} + Rect_{IV}) = 2.82+4+4.90+5.66=17.38.

Thus the area under the curve is approximately 17.38.

### Trapezoidal Approximations

Another method of approximating the area under a curve is to use trapezoids. See Figure 12.2-5.

#### Formula for Trapezoidal Approximation

If *f* is continuous, the area under the curve of *f* from *x* = *a* to *x* = *b* is:

#### Example 1

Find the approximate area under the curve of from *x* = 0 to *x* = π, using 4 trapezoids. (See Figure 12.2-6.)

Since *n* = 4,

Area under the curve:

Practice problems for these concepts can be found at: Areas and Volumes Practice Problems for AP Calculus

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