Math / Calculus

QuestionUse the midpoint rule with the given value of $n$ to approximate the integral. $\int_{0}^{3} 3 e^{\sqrt{x}} d x, \quad n=6$
Find the width of each subinterval. $\square$ 0.5 - units
Find the midpoints of the subintervals where $x_{1}<x_{2}<\ldots<x_{6}$ . $\begin{array}{l} \bar{x}_{1}=0.25 \\ \bar{x}_{2}=75 \\ \bar{x}_{3}=1.25 \\ \bar{x}_{4}=1.75 \\ \bar{x}_{5}=2.25 \\ \bar{x}_{6}=2.75 \end{array}$
Approximate the integral. Round the answer to four decimal places. $M_{6}=\square \times$ .

Studdy Solution

STEP 1

1. We are using the midpoint rule to approximate the integral.
2. The integral is $\int_{0}^{3} 3 e^{\sqrt{x}} \, dx$ .
3. The number of subintervals is $n = 6$ .

STEP 2

1. Calculate the width of each subinterval.
2. Determine the midpoints of each subinterval.
3. Apply the midpoint rule to approximate the integral.

STEP 3

Calculate the width of each subinterval:
The formula for the width $\Delta x$ of each subinterval is:
$\Delta x = \frac{b-a}{n}$
where $a = 0$ , $b = 3$ , and $n = 6$ .
Substitute the values:
$\Delta x = \frac{3-0}{6} = 0.5$

STEP 4

Determine the midpoints of each subinterval:
The midpoints are given as:
\begin{align} \bar{x}_{1} &= 0.25 \\ \bar{x}_{2} &= 0.75 \\ \bar{x}_{3} &= 1.25 \\ \bar{x}_{4} &= 1.75 \\ \bar{x}_{5} &= 2.25 \\ \bar{x}_{6} &= 2.75 \\ \end{align}

STEP 5

Apply the midpoint rule to approximate the integral:
The midpoint rule formula is:
$M_n = \Delta x \sum_{i=1}^{n} f(\bar{x}_i)$
Substitute $\Delta x = 0.5$ and $f(x) = 3e^{\sqrt{x}}$ :
$M_6 = 0.5 \left( 3e^{\sqrt{0.25}} + 3e^{\sqrt{0.75}} + 3e^{\sqrt{1.25}} + 3e^{\sqrt{1.75}} + 3e^{\sqrt{2.25}} + 3e^{\sqrt{2.75}} \right)$
Calculate each term:
\begin{align} f(\bar{x}_1) &= 3e^{\sqrt{0.25}} = 3e^{0.5} \\ f(\bar{x}_2) &= 3e^{\sqrt{0.75}} \\ f(\bar{x}_3) &= 3e^{\sqrt{1.25}} \\ f(\bar{x}_4) &= 3e^{\sqrt{1.75}} \\ f(\bar{x}_5) &= 3e^{\sqrt{2.25}} \\ f(\bar{x}_6) &= 3e^{\sqrt{2.75}} \\ \end{align}
Substitute these back into the equation:
$M_6 = 0.5 \left( 3e^{0.5} + 3e^{\sqrt{0.75}} + 3e^{\sqrt{1.25}} + 3e^{\sqrt{1.75}} + 3e^{\sqrt{2.25}} + 3e^{\sqrt{2.75}} \right)$
Calculate the sum and multiply by $0.5$ to get the approximate value of the integral.
The approximate value of the integral is:
$M_6 \approx 16.4377$

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