Riemann Sums Calculator

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Example

Created on 2024-06-20Asked by Mila Taylor (Solvelet student)

Use a right Riemann sum with

n = 4

to approximate the integral

\int_{0}^{2} (4 - x^2) \, dx.

Solution

To approximate

\int_{0}^{2} (4 - x^2) \, dx

using a right Riemann sum with

n = 4

: 1. **Divide the interval

[0, 2]

into

n = 4

subintervals of equal width:**

\Delta x = \frac{2 - 0}{4} = 0.5.

2. **Determine the right endpoints of each subinterval:**

x_1 = 0.5, \quad x_2 = 1.0, \quad x_3 = 1.5, \quad x_4 = 2.0.

3. **Evaluate the function at each right endpoint:**

f(x_1) = 4 - (0.5)^2 = 4 - 0.25 = 3.75,

f(x_2) = 4 - (1.0)^2 = 4 - 1 = 3,

f(x_3) = 4 - (1.5)^2 = 4 - 2.25 = 1.75,

f(x_4) = 4 - (2.0)^2 = 4 - 4 = 0.

4. **Calculate the right Riemann sum:**

R_4 = \Delta x [f(x_1) + f(x_2) + f(x_3) + f(x_4)],

R_4 = 0.5 [3.75 + 3 + 1.75 + 0] = 0.5 \cdot 8.5 = 4.25.

Therefore, the approximate value of the integral using a right Riemann sum with

n = 4

4.25

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Harper Lewis on Solvelet

1. Estimate the area under the curve

y = x^2

on the interval

[0, 2]

using four rectangles and right endpoints.2. Use a Riemann sum to approximate the area under the curve

y = \sqrt{x}

on the interval

[0, 1]

using six subintervals of equal width.,

DefinitionIn calculus, Riemann sums are methods for approximating the total area under a curve on a graph, otherwise known as the integral of a function. They represent the sum of rectangles or trapezoids that estimate the area. Example: The left Riemann sum is an approximation of ∫abf(x)dx by adding the areas of f(xi)Δx where i=0,1,...,n−1, and Δx=nb−a.

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