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

Estimate the area between the graph of

(

)

cos

(

)

and the

-axis

on the interval

–

Use 4 trapezoids of equal width to find

What is the value of

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

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Learn with an example

question

Estimate the area between the graph of

(

)

–

and the

-axis

on the interval

[

–

]

Use 4 trapezoids of equal width 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

trapezoids that have width

and parallel sides of height

(

–

)

and

(

)

where

–

and

are the endpoints of the

ith

subinterval,

[

–

]

So, the area of the

ith

trapezoid is

(

–

)

(

)

This is called the trapezoidal

approximation,

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

(

–

)

(

)

(

)

(

)

(

)

(

)

...

(

–

)

(

)

[

(

)

(

)

(

)

...

(

–

)

(

)

]

solution

To estimate the area between the graph of

(

)

–

and the

-axis

on the interval

[

–

]

you can use 4 trapezoids of equal width.

Since you want to use 4 trapezoids, first partition the interval

[

–

]

into 4 subintervals of equal width

–

(

–

)

These are the subintervals.

[

–

]

[

–

]

[

]

[

]

So, the endpoints of the subintervals are

–

and

For each trapezoid, the heights of the parallel sides are equal to the value of the function at the corresponding endpoints, and the width is

For example, the heights of the parallel sides of the second trapezoid are

(

–

)

and

(

)

and the width is 1. So, the area of the second trapezoid is

(

–

)

(

)

(

)

This is the sum of the areas of the 4 trapezoids.

(

–

)

(

)

[

(

)

(

)

(

)

(

)

(

)

]

[

(

–

)

(

–

)

(

)

(

)

(

)

]

So, you can estimate the area between

(

)

–

and the x-axis on the interval

[

–

]

using

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

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