complete the following steps for the given function, interval, and value of n. a. sketch the graph of the…

complete the following steps for the given function, interval, and value of n. a. sketch the graph of the function on the given interval. b. calculate δx and the grid points x0, x1, …, xn. c. illustrate the left and right riemann sums, and determine which riemann sum underestimates and which sum overestimates the area under the curve. d. calculate the left and right riemann sums. f(x)=2x² + 5 on 2,7; n = 5 the right riemann sum is . (simplify your answer.)

complete the following steps for the given function, interval, and value of n. a. sketch the graph of the function on the given interval. b. calculate δx and the grid points x0, x1, …, xn. c. illustrate the left and right riemann sums, and determine which riemann sum underestimates and which sum overestimates the area under the curve. d. calculate the left and right riemann sums. f(x)=2x² + 5 on 2,7; n = 5 the right riemann sum is . (simplify your answer.)

Answer

Explanation:

Step1: Calculate $\Delta x$

The formula for $\Delta x=\frac{b - a}{n}$, where $a = 2$, $b = 7$ and $n=5$. So $\Delta x=\frac{7 - 2}{5}=1$.

Step2: Find the grid - points

The grid - points $x_i=a + i\Delta x$. So $x_0 = 2$, $x_1=2 + 1=3$, $x_2=2+2\times1 = 4$, $x_3=2 + 3\times1=5$, $x_4=2+4\times1 = 6$, $x_5=2+5\times1=7$.

Step3: Calculate the right - Riemann sum

The right - Riemann sum $R_n=\sum_{i = 1}^{n}f(x_i)\Delta x$. Here, $\Delta x = 1$ and $f(x)=2x^{2}+5$. [ \begin{align*} R_5&=f(x_1)+f(x_2)+f(x_3)+f(x_4)+f(x_5)\ &=(2\times3^{2}+5)+(2\times4^{2}+5)+(2\times5^{2}+5)+(2\times6^{2}+5)+(2\times7^{2}+5)\ &=(2\times9 + 5)+(2\times16+5)+(2\times25+5)+(2\times36+5)+(2\times49+5)\ &=(18 + 5)+(32+5)+(50+5)+(72+5)+(98+5)\ &=23+37+55+77+103\ &=295 \end{align*} ]

Answer:

$295$