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AP Calculus AB. Day 5 Section 4.2. Area Under a Curve. Find the area of the region bounded by y = f(x), the x-axis, x = a, and x = b. Approximate the area by creating rectangles of equal width whose endpoints are on f(x). n = # of rectangles. Each left endpoint is on f(x).

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AP Calculus AB

Day 5

Section 4.2

Perkins

Area Under a Curve

Find the area of the region bounded by y = f(x), the x-axis, x = a, and x = b.

Approximate the area by creating rectangles of equal width whose endpoints are on f(x).

n = # of rectangles

Each left endpoint is on f(x)

Each right endpoint is on f(x)

This over-estimates the area under the curve…

This under-estimates the area under the curve…

We call it the Upper Sum.

We call it the Lower Sum.

Each method is called a Riemann Sum.

How do we make these approximations for the area under a curve more accurate?

Use more rectangles.

(Always choose whichever sum involves right endpoints.)

The Limit Definition for finding the area under a curve:

or

Find the area beneath (above the x-axis) in the interval [1,3].

a. Use 1 rectangle.

b. Use 2 rectangles.

If a specific number of rectangles is given, it is often easier to find the area without using sigma!

Find the area beneath (above the x-axis) in the interval [1,3].

c. Use the limit definition.

AP Calculus AB

Day 5

Section 4.2

Perkins

Area Under a Curve

Find the area of the region bounded by y = f(x), the x-axis, x = a, and x = b.

Approximate the area by creating rectangles of equal width whose endpoints are on f(x).

How do we make these approximations for the area under a curve more accurate?

The Limit Definition for finding the area under a curve:

Find the area beneath (above the x-axis) in the interval [1,3].

a. Use 1 rectangle.

b. Use 2 rectangles.

Find the area beneath (above the x-axis) in the interval [1,3].

c. Use the limit definition.