IXL - Relate Riemann sums and definite integrals (Calculus practice)

Express the definite integral as a limit of Riemann sums that use right endpoints.

∫

(

)

Which limit is the result?

lim

→∞

lim

→∞

lim

→∞

lim

→∞

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Learn with an example

question

Express the definite integral as a limit of Riemann sums that use right endpoints.

∫

Which limit is the result?

lim

→∞

lim

→∞

lim

→∞

lim

→∞

key idea

Let

(

)

be a function defined on the interval

[

]

A Riemann sum approximates the area between the graph of

(

)

and the

-axis

using rectangles.

To set up a Riemann sum, divide the interval

[

]

into

subintervals of equal width

–

Choose a point

in each subinterval. Each rectangle has width

and height

(

)

So, the area under the curve can be approximated as

(

)

The definite integral of

(

)

[

]

represents the exact area between the graph of

(

)

and the

-axis.

It is written

∫

(

)

and, if it exists, equals a limit of Riemann sums as

the number of rectangles, approaches infinity.

∫

(

)

lim

→∞

(

)

For example, you can choose

to be the right endpoint of each subinterval. In that case,

and the formula becomes

∫

(

)

lim

→∞

(

)

solution

Write

∫

as a limit of Riemann sums using right endpoints. To do this, divide the interval

[

]

into

subintervals. You need to find

and

First, find

–

Substitute

and

–

Substitute

and

Next, find

Substitute

and

Substitute

and

Now, use the formula to write the integral of

(

)

as a limit of Riemann sums.

∫

(

)

lim

→∞

(

)

∫

lim

→∞

lim

→∞

$Confetti$

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