Question

Calculate the left and right Riemann sums, for n=3 on the interval [1,7]:


f(x)=2x^2


Group of answer choices:


a) Left: 182. Right: 278.

b) Left: 70: Right: 166.

c) Left: 364. Right: 556.

d) Left: 140. Right: 332.

Answer 1

Left Riemann and Right Riemann sums of $f(x) = 2\cdot x^{2}$ for $n = 3$ on the interval $[1,7]$ are 140 and 332, respectively. (Correct choice: D)

Riemann sums are approximation-based methods to estimate the area below a curve by using rectangles. There two forms of Riemann sums:

Left Riemann sum

$A = \frac{b-a}{n} \cdot \Sigma\limits_{k= 0}^{n-1}f(a+\frac{k\cdot b}{n}-\frac{k\cdot a}{n})$ (1)

Right Riemann sum

$A = \frac{b-a}{n} \cdot \Sigma\limits_{k= 1}^{n} f(a+\frac{(k+1)\cdot b}{n}-\frac{(k+1)\cdot a}{n} )$ (2)

Where:

$a$ - Lower bound.

$b$ - Upper bound.

$n$ - Number of rectangles.

$k$ - Index.

$A$ - Estimated area.

If we know that $f(x) = 2\cdot x^{2}$, $a = 1$, $b = 7$ and $n = 3$, then the Riemann sums are:

Left Riemann sum

$A = \frac{7-1}{3} \cdot [f(1) + f(3) + f(5)]$

$A = 2\cdot (2+18+50)$

$A = 140$

Right Riemann sum

$A = \frac{7-1}{3}\cdot [f(3)+f(5)+f(7)]$

$A = 2\cdot (18 + 50 +98)$

$A = 332$

Left Riemann and Right Riemann sums of $f(x) = 2\cdot x^{2}$ for $n = 3$ on the interval $[1,7]$ are 140 and 332, respectively. (Correct choice: D)

We kindly invite to see this question on Riemann sums: https://brainly.com/question/2759901