IXL - Approximate the area under a curve: midpoints (Calculus practice)

Estimate the area between the graph of

(

)

–

and the

-axis

on the interval

[

–

]

Use 4 rectangles of equal width and midpoints to find

What is the value of

Write your answer as a decimal rounded to the nearest hundredth.

ref_doc_title.

Learn with an example

question

Estimate the area between the graph of

(

)

and the

-axis

on the interval

[

]

Use 5 rectangles of equal width and midpoints to find

What is the value of

Write your answer as a decimal rounded to the nearest hundredth.

key idea

Let

(

)

be a function defined on the interval

[

]

Partition the interval

[

]

into

subintervals

[

]

[

]

, ...,

[

–

]

where

. . .

–

and each subinterval has equal width

–

The area between the graph of

(

)

and the

-axis

can be approximated using

rectangles of width

and height

(

)

where

is the midpoint of the

subinterval,

[

–

]

–

This is called the midpoint

approximation,

and the approximation for the area is given by the sum of the areas of the rectangles,

(

)

(

)

(

)

...

(

–

)

(

)

...

–

solution

To estimate the area between the graph of

(

)

and the

-axis

on the interval

[

]

you can use 5 rectangles of equal width and midpoints.

Since you want to use 5 rectangles, first partition the interval

[

]

into 5 subintervals of equal width

–

These are the subintervals.

[

]

[

]

[

]

[

]

[

]

So, the midpoints of the subintervals are

and

The height of each rectangle is equal to the value of the function at the corresponding midpoint, and the width of each rectangle is

For example, the height of the second rectangle is

and the width is 1.

This is the sum of the areas of the 5 rectangles.

(

)

(

)

(

)

(

)

(

)

(

)

13.8305…

So, rounded to the nearest hundredth, you can estimate the area between the graph of

(

)

and the

-axis

on the interval

[

]

using

≈

13.83

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Q.1Approximate the area under a curve: left and right endpoints

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