right end point approximation using rectangles

Question

approximate the area under the curve  from to using a Right Endpoint approximation with 4 subdivisions.

Answer 1

Hi Kaylee,

It might be a little difficult to explain this clearly without being able to use a picture, but let's give it a try!

We are going to estimate the area under the curve by drawing 4 rectangles, finding the area of each one, and then adding the four areas together. Remember that the area of a rectangle is

A = length x width; or

A = base x height; or

A = bh

If we are going to divide the area under the parabola y = x² from x = 3 to x = 5 into four rectangles of equal width, then each rectangle will have the width (5 - 3)/4 = 2/4 = 1/2. The 5-3 calculates the width of the whole interval and we divide that into 4 pieces.

So, the first rectangle spans from 3 to 3.5.

The second rectangle spans from 3.5 to 4.

The third rectangle spans from 4 to 4.5.

The last rectangle spans from 4.5 to 5.

Notice the right sides of each rectangle have x values equal to 3.5, 4, 4.5, and 5.

We will use these values to calculate the height of our 4 rectangles.

So, for the first rectangle, the base is equal to 1/2.

The height is given by the function value, or the y value, when x = 3.5 (the right endpoint of the first rectangle). Since y = x², the height of the first rectangle is y = 3.5² = 12.25.

We calculate the area of the other three rectangles similarly. Remember that the base of each rectangle is 1/2. The height of each rectangle is the y value at the right endpoint. Therefore, if we add ALL FOUR rectangles together, we get

A = 0.5 (3.5)² + 0.5 (4)² + 0.5 (4.5)² + 0.5 (5)²

A = 0.5 (12.25) + 0.5 (16) + 0.5 (20.25) + 0.5 (25)

A = 6.125 + 8 + 10.125 + 12.5

A = 36.75

Hope this helps!

JMJ