Area Under a Curve - Pre-Calculus
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Approximate the area under the following curve from 
to 
using a midpoint Riemann sum with 
:
Approximate the area under the following curve from
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If we want to approximate the area under a curve using n=4, that means we will be using 4 rectangles. Because the problem asks us to approximate the area from x=0 to x=4, this means we will have a rectangle between x=0 and x=1, between x=1 and x=2, between x=2 and x=3, and between x=3 and x=4. We want to use a midpoint Riemann sum, so the height of each rectangle will be the value of the function at the midpoint of each interval:
Now that we know the height of each rectangle, all we have to do is find its area by multiplying the height by the width, which is just 1 for each rectangle. Then we add the area of all the rectangles to find our approximation for the area under the curve from x=0 to x=4:
If we want to approximate the area under a curve using n=4, that means we will be using 4 rectangles. Because the problem asks us to approximate the area from x=0 to x=4, this means we will have a rectangle between x=0 and x=1, between x=1 and x=2, between x=2 and x=3, and between x=3 and x=4. We want to use a midpoint Riemann sum, so the height of each rectangle will be the value of the function at the midpoint of each interval:
Now that we know the height of each rectangle, all we have to do is find its area by multiplying the height by the width, which is just 1 for each rectangle. Then we add the area of all the rectangles to find our approximation for the area under the curve from x=0 to x=4:
Using a left hand approximation and 
rectangles, what is the area under the curve 
on the interval 
?
Using a left hand approximation and
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A left hand approximation requires us to slice the region into four rectangles of equal width. Since the interval on which we are slicing is 
and we want to create four rectangles of equal width, the width of each rectangle must be,
.
The height of each rectangular slice is given by the function value at the left edge of each rectangle. Beginning at the leftmost edge of the first rectangle on the interval, these left endpoints exist at 
, but not at 
as that is the right edge of the fourth and final rectangle on the interval.
Using the function given, 
, the first rectangle has height 
. The rest have height 
.
Multiply each by the width of each rectangle 
to get the area of each rectangular slice. Then add the area of the slices together, and simplify to get the correct result.
![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/362727/gif.latex "\ Area =\sum_${k=0}^{3}$f(x_k)\Delta x \ \= f(0)\Delta x+f(0.5)\Delta x+f(1.0)\Delta x+f(1.5)\Delta x\ \=[f(0)+f(0.5)+f(1.0)+f(1.5)]\Delta x\")
A left hand approximation requires us to slice the region into four rectangles of equal width. Since the interval on which we are slicing is
The height of each rectangular slice is given by the function value at the left edge of each rectangle. Beginning at the leftmost edge of the first rectangle on the interval, these left endpoints exist at
Using the function given,
Multiply each by the width of each rectangle
Let 
Approximate the area underneath the function on the interval 
divided into four sub-intervals using the midpoint height of a rectangle for each sub-interval.
Let
Approximate the area underneath the function on the interval
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The interval 
divided into four sub-intervals gives rectangles with vertices of the bases at
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or 
, 
, 
, and 
Because each sub-interval has width 
, the approximated area using rectangles is
As such, the approximated area is
units squared
The interval
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or
Because each sub-interval has width
As such, the approximated area is
Let 
Approximate the area underneath the function on the interval 
divided into two sub-intervals using the midpoint height of a rectangle for each sub-interval.
Let
Approximate the area underneath the function on the interval
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The interval 
divided into two sub-intervals gives two rectangles with vertices of the bases at
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or 
and 
Because each sub-interval has width 
, the approximated area using rectangles is
As such, the approximated area is
units squared
The interval
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or
Because each sub-interval has width
As such, the approximated area is
Approximate the area under the curve given by 
on the interval 
using left endpoints.
Approximate the area under the curve given by
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In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the left endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 100.
In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the left endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 100.
Approximate the area under the curve given by 
on the interval 
using right endpoints.
Approximate the area under the curve given by
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In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the right endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 225.
In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the right endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 225.
What is the area under the curve of the function
from 
to 
.
What is the area under the curve of the function
from
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The area under the curve of the function 
is the definite integral from 
to 
.
Remember when integrating, we will increase the exponent by one and then divide the whole term by the value of the new exponent.
From here, we find the difference between the function values of the boundaries.
The area under the curve of the function
Remember when integrating, we will increase the exponent by one and then divide the whole term by the value of the new exponent.
From here, we find the difference between the function values of the boundaries.
Approximate the area under the following curve from to using a midpoint Riemann sum with :
Approximate the area under the following curve from
Tap to see back →
If we want to approximate the area under a curve using n=4, that means we will be using 4 rectangles. Because the problem asks us to approximate the area from x=0 to x=4, this means we will have a rectangle between x=0 and x=1, between x=1 and x=2, between x=2 and x=3, and between x=3 and x=4. We want to use a midpoint Riemann sum, so the height of each rectangle will be the value of the function at the midpoint of each interval:
Now that we know the height of each rectangle, all we have to do is find its area by multiplying the height by the width, which is just 1 for each rectangle. Then we add the area of all the rectangles to find our approximation for the area under the curve from x=0 to x=4:
If we want to approximate the area under a curve using n=4, that means we will be using 4 rectangles. Because the problem asks us to approximate the area from x=0 to x=4, this means we will have a rectangle between x=0 and x=1, between x=1 and x=2, between x=2 and x=3, and between x=3 and x=4. We want to use a midpoint Riemann sum, so the height of each rectangle will be the value of the function at the midpoint of each interval:
Now that we know the height of each rectangle, all we have to do is find its area by multiplying the height by the width, which is just 1 for each rectangle. Then we add the area of all the rectangles to find our approximation for the area under the curve from x=0 to x=4:
Using a left hand approximation and rectangles, what is the area under the curve on the interval ?
Using a left hand approximation and
Tap to see back →
A left hand approximation requires us to slice the region into four rectangles of equal width. Since the interval on which we are slicing is and we want to create four rectangles of equal width, the width of each rectangle must be,
.
The height of each rectangular slice is given by the function value at the left edge of each rectangle. Beginning at the leftmost edge of the first rectangle on the interval, these left endpoints exist at , but not at as that is the right edge of the fourth and final rectangle on the interval.
Using the function given, , the first rectangle has height . The rest have height .
Multiply each by the width of each rectangle to get the area of each rectangular slice. Then add the area of the slices together, and simplify to get the correct result.
A left hand approximation requires us to slice the region into four rectangles of equal width. Since the interval on which we are slicing is
The height of each rectangular slice is given by the function value at the left edge of each rectangle. Beginning at the leftmost edge of the first rectangle on the interval, these left endpoints exist at
Using the function given,
Multiply each by the width of each rectangle
Let
Approximate the area underneath the function on the interval divided into four sub-intervals using the midpoint height of a rectangle for each sub-interval.
Let
Approximate the area underneath the function on the interval
Tap to see back →
The interval divided into four sub-intervals gives rectangles with vertices of the bases at
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or , , , and
Because each sub-interval has width , the approximated area using rectangles is
As such, the approximated area is
units squared
The interval
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or
Because each sub-interval has width
As such, the approximated area is
Let
Approximate the area underneath the function on the interval divided into two sub-intervals using the midpoint height of a rectangle for each sub-interval.
Let
Approximate the area underneath the function on the interval
Tap to see back →
The interval divided into two sub-intervals gives two rectangles with vertices of the bases at
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or and
Because each sub-interval has width , the approximated area using rectangles is
As such, the approximated area is
units squared
The interval
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or
Because each sub-interval has width
As such, the approximated area is
Approximate the area under the curve given by on the interval using left endpoints.
Approximate the area under the curve given by
Tap to see back →
In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the left endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 100.
In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the left endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 100.
Approximate the area under the curve given by on the interval using right endpoints.
Approximate the area under the curve given by
Tap to see back →
In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the right endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 225.
In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the right endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 225.
What is the area under the curve of the function
from to .
What is the area under the curve of the function
from
Tap to see back →
The area under the curve of the function is the definite integral from to .
Remember when integrating, we will increase the exponent by one and then divide the whole term by the value of the new exponent.
From here, we find the difference between the function values of the boundaries.
The area under the curve of the function
Remember when integrating, we will increase the exponent by one and then divide the whole term by the value of the new exponent.
From here, we find the difference between the function values of the boundaries.
Approximate the area under the following curve from to using a midpoint Riemann sum with :
Approximate the area under the following curve from
Tap to see back →
If we want to approximate the area under a curve using n=4, that means we will be using 4 rectangles. Because the problem asks us to approximate the area from x=0 to x=4, this means we will have a rectangle between x=0 and x=1, between x=1 and x=2, between x=2 and x=3, and between x=3 and x=4. We want to use a midpoint Riemann sum, so the height of each rectangle will be the value of the function at the midpoint of each interval:
Now that we know the height of each rectangle, all we have to do is find its area by multiplying the height by the width, which is just 1 for each rectangle. Then we add the area of all the rectangles to find our approximation for the area under the curve from x=0 to x=4:
If we want to approximate the area under a curve using n=4, that means we will be using 4 rectangles. Because the problem asks us to approximate the area from x=0 to x=4, this means we will have a rectangle between x=0 and x=1, between x=1 and x=2, between x=2 and x=3, and between x=3 and x=4. We want to use a midpoint Riemann sum, so the height of each rectangle will be the value of the function at the midpoint of each interval:
Now that we know the height of each rectangle, all we have to do is find its area by multiplying the height by the width, which is just 1 for each rectangle. Then we add the area of all the rectangles to find our approximation for the area under the curve from x=0 to x=4:
Using a left hand approximation and rectangles, what is the area under the curve on the interval ?
Using a left hand approximation and
Tap to see back →
A left hand approximation requires us to slice the region into four rectangles of equal width. Since the interval on which we are slicing is and we want to create four rectangles of equal width, the width of each rectangle must be,
.
The height of each rectangular slice is given by the function value at the left edge of each rectangle. Beginning at the leftmost edge of the first rectangle on the interval, these left endpoints exist at , but not at as that is the right edge of the fourth and final rectangle on the interval.
Using the function given, , the first rectangle has height . The rest have height .
Multiply each by the width of each rectangle to get the area of each rectangular slice. Then add the area of the slices together, and simplify to get the correct result.
A left hand approximation requires us to slice the region into four rectangles of equal width. Since the interval on which we are slicing is
The height of each rectangular slice is given by the function value at the left edge of each rectangle. Beginning at the leftmost edge of the first rectangle on the interval, these left endpoints exist at
Using the function given,
Multiply each by the width of each rectangle
Let
Approximate the area underneath the function on the interval divided into four sub-intervals using the midpoint height of a rectangle for each sub-interval.
Let
Approximate the area underneath the function on the interval
Tap to see back →
The interval divided into four sub-intervals gives rectangles with vertices of the bases at
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or , , , and
Because each sub-interval has width , the approximated area using rectangles is
As such, the approximated area is
units squared
The interval
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or
Because each sub-interval has width
As such, the approximated area is
Let
Approximate the area underneath the function on the interval divided into two sub-intervals using the midpoint height of a rectangle for each sub-interval.
Let
Approximate the area underneath the function on the interval
Tap to see back →
The interval divided into two sub-intervals gives two rectangles with vertices of the bases at
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or and
Because each sub-interval has width , the approximated area using rectangles is
As such, the approximated area is
units squared
The interval
For the approximated area, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or
Because each sub-interval has width
As such, the approximated area is
Approximate the area under the curve given by on the interval using left endpoints.
Approximate the area under the curve given by
Tap to see back →
In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the left endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 100.
In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the left endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 100.
Approximate the area under the curve given by on the interval using right endpoints.
Approximate the area under the curve given by
Tap to see back →
In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the right endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 225.
In order to approximate the area under a curve using rectangles, one must take the sum of the areas of discrete rectangles under the curve. Taking the height of each rectangle as the function evaluated at the right endpoint, we obtain the following rectangle areas:
The sum of the individual rectangles yields an overall area approximation of 225.