# Find the riemann sum for #f(x)=x+x^2#?

A Riemann sum is a certain type of approximations for an integral. The easiest and simplest method to use is a rectangle sum; where we sum the area of

Here's a Riemann sum with

This is just an example. Instead, consider

Let

Now; imagine

Let

Another property about our x-axis marks is that the lenght/non-width side of the i-th rectangle is going to be

Since we want the area until

The areas of the rectangles approximate the areas under the graph of

One way to make

By this definition,

The more rectangles we have, the better the sum approximates the integral. Thus, we can say that

Generalisation: Since we want the area from

This is one of the many Riemann Sum formulae for a function

In our case,

if there are

By signing up, you agree to our Terms of Service and Privacy Policy

To find the Riemann sum for ( f(x) = x + x^2 ), you need to first specify the interval over which you want to compute the Riemann sum and the method of partitioning that interval. Let's assume we want to find the Riemann sum for ( f(x) = x + x^2 ) over the interval ([a, b]), and we'll use ( n ) subintervals of equal width.

The width of each subinterval will be ( \Delta x = \frac{b - a}{n} ). Then, the endpoints of the subintervals are ( x_0 = a, x_1, x_2, ..., x_{n-1}, x_n = b ), where ( x_i = a + i\Delta x ) for ( i = 0, 1, ..., n ).

Now, for each subinterval ( [x_{i-1}, x_i] ), choose a sample point ( c_i ) within that interval. The Riemann sum is then given by:

[ \sum_{i=1}^{n} f(c_i) \Delta x ]

where ( f(c_i) ) is the value of the function ( f(x) ) evaluated at the sample point ( c_i ) within the ( i )th subinterval.

Therefore, the Riemann sum for ( f(x) = x + x^2 ) over the interval ([a, b]) with ( n ) subintervals of equal width is:

[ \sum_{i=1}^{n} (c_i + c_i^2) \Delta x ]

where ( \Delta x = \frac{b - a}{n} ), and ( c_i ) is any point in the ( i )th subinterval.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you use Riemann sums to evaluate the area under the curve of #f(x) = (e^x) − 5# on the closed interval [0,2], with n=4 rectangles using midpoints?
- How do you determine the area enclosed by an ellipse #x^2/5 + y^2/ 3# using the trapezoidal rule?
- How to you approximate the integral of # (t^3 +t) dx# from [0,2] by using the trapezoid rule with n=4?
- How do you use the Trapezoidal rule and three subintervals to give an estimate for the area between #y=cscx# and the x-axis from #x= pi/8# to #x = 7pi/8#?
- What is Integration Using the Trapezoidal Rule?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7