Question

Use the riemann sum with n=2 rectangles to approximate f10 1/x4 + 1 dx . Round to 3 decimal places.

Answer 1

Answer:

0.971 it's the right answer thankyou

Answer 2

Answer:

Reimann sum of given function f(x) = $\int\limits^1_0 {\frac{1}{x^{4}+1 } } \, dx$= 0.8455 and by rounding off to 3 decimal places we get f(x) = 0.846.

Concept:

A regions approximate area, known as a Riemann sum, is calculated by summing the areas of various simplified slices of the region. Calculus uses it to formalize the process of exhaustion, which is used to calculate a region's area. The result of this operation is the integral, which precisely calculates the value of the area.

Given:

The given function f(x) = $\int\limits^1_0 {\frac{1}{x^{4}+1 } } \, dx$.

Find:

To find the Riemann sum with n = 2 rectangles to approximate f(x) = $\int\limits^1_0 {\frac{1}{x^{4}+1 } } \, dx$.

Solution:

We take into account the values at the right ends for lesser sums and the values at the left ends for higher sums.

$\Delta x = \frac{b-a}{n} \\\Delta x = \frac{1-0}{2}\\\Delta x = 0.5$

Now, for lower rectangle sum, $A_{L}$ = $\Delta x |f(0) + f(0.5) |$

$A_{L} =0.5 |f(0) + f(0.5) |$

$A_{L} = 0.5(1 + \frac{16}{17} )\\A_{L} = 0.9705$

Again for upper rectangle sum, $A_{R} = \Delta x |f(0) + f(1) |$

$A_{R}= \Delta x |f(0) + f(1) |\\A_{R} = 0.5[\frac{1}{2} + \frac{1}{1 + \frac{1}{16} } ]\\A_{R} = 0.7205$

So, A = $\frac{A_{L}+A_{R} }{2}$

A = $\frac{0.9705+ 0.7205}{2}$

A = 0.8455

With Reimann sum we will get f(x) = $\int\limits^1_0 {\frac{1}{x^{4}+1 } } \, dx$= 0.8455

Hence, Reimann sum we will get f(x) = $\int\limits^1_0 {\frac{1}{x^{4}+1 } } \, dx$= 0.8455. Now, for round off to 3 decimal places we get f(x) = 0.846.

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