How do i answer this? # int_2^4 \ (2x)/(x^2+1) # via a Riemann sum.

Question

Evelyn Agard · Accepted Answer

# A = lim_(n rarr oo) \ sum_(i=1)^n 2/n \ (2((2n+2i)/n)) / (( (2n+2i)^2/n^2 + 1) #
# \ \ \ = lim_(n rarr oo) \ sum_(i=1)^n (4(2n+2i)) / ( (2n+2i)^2+n^2) #

We seek: # int_2^4 \ (2x)/(x^2+1) # via a Riemann sum. So, we have: # A = lim_(n rarr oo) \ sum_(i=1)^n f(x_i) Delta x # # \ \ \ = lim_(n rarr oo) \ sum_(i=1)^n f(a+iDelta x) Delta x # # \ \ \ = lim_(n rarr oo) \ sum_(i=1)^n f(2+((4-2)i)/n) (4-2)/n # # \ \ \ = lim_(n rarr oo) \ sum_(i=1)^n f(2+(2i)/n) (2)/n # # \ \ \ = lim_(n rarr oo) \ sum_(i=1)^n 2/n \ f( (2n+2i )/n ) # Then using the definition of #f(x)=(2x)/(x^2+1)# this becomes: # A = lim_(n rarr oo) \ sum_(i=1)^n 2/n \ (2((2n+2i)/n))/(((2n+2i)/n)^2+1) # for a 1 mark question this answer is probably sufficient, but it could be simplified further if required: # A = lim_(n rarr oo) \ sum_(i=1)^n 2/n \ (2((2n+2i)/n)) / (( (2n+2i)^2/n^2 + n^2/n^2) # # \ \ \ = lim_(n rarr oo) \ sum_(i=1)^n (4((2n+2i)/n^2)) / (( (2n+2i)^2+n^2)/n^2) # # \ \ \ = lim_(n rarr oo) \ sum_(i=1)^n (4(2n+2i)) / ( (2n+2i)^2+n^2) #

Adam Adkins · Answer

undefined To answer the integral $\int_{2}^{4} \frac{2x}{x^2 + 1}$ via a Riemann sum, you first partition the interval $[2, 4]$ into $n$ subintervals of equal width $\Delta x$. The width of each subinterval is $\Delta x = \frac{4 - 2}{n} = \frac{2}{n}$. Then, you choose sample points $x_i^*$ in each subinterval, typically the right endpoint of each subinterval. The Riemann sum is given by:

$$
\sum_{i=1}^{n} f(x_i^*) \Delta x
$$

where $f(x)$ is the function being integrated.

For this integral, $f(x) = \frac{2x}{x^2 + 1}$. So, the Riemann sum becomes:

$$
\sum_{i=1}^{n} \frac{2x_i^*}{(x_i^*)^2 + 1} \Delta x
$$

You then take the limit as $n$ approaches infinity to obtain the definite integral:

$$
\int_{2}^{4} \frac{2x}{x^2 + 1} \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} \frac{2x_i^*}{(x_i^*)^2 + 1} \Delta x
$$

This limit represents the area under the curve $\frac{2x}{x^2 + 1}$ on the interval $[2, 4]$ using Riemann sums.