Math / Calculus

QuestionUse the Midpoint Rule with $n=4$ to approximate the value of the definite integral. Use a graphing utility to verify your result. (Round your answer to three decimal places.) $\int_{3}^{5} \frac{18}{x} d x$

Studdy Solution

STEP 1

1. We are asked to approximate the definite integral $\int_{3}^{5} \frac{18}{x} \, dx$ using the Midpoint Rule.
2. We will use $n = 4$ subintervals for the approximation.
3. We will round the final answer to three decimal places.
4. A graphing utility will be used to verify the result.

STEP 2

1. Determine the width of each subinterval.
2. Identify the midpoints of each subinterval.
3. Evaluate the function at each midpoint.
4. Apply the Midpoint Rule to approximate the integral.
5. Verify the result using a graphing utility.

STEP 3

Determine the width of each subinterval:
The interval $[3, 5]$ is divided into $n = 4$ subintervals. The width $\Delta x$ of each subinterval is given by:
$\Delta x = \frac{b-a}{n} = \frac{5-3}{4} = \frac{2}{4} = 0.5$

STEP 4

Identify the midpoints of each subinterval:
The subintervals are $[3, 3.5]$ , $[3.5, 4]$ , $[4, 4.5]$ , and $[4.5, 5]$ . The midpoints $x_i^$ are:
$x_1^ = 3.25, \quad x_2^* = 3.75, \quad x_3^* = 4.25, \quad x_4^* = 4.75$

STEP 5

Evaluate the function at each midpoint:
$f(x) = \frac{18}{x}$
Calculate $f(x_i^*)$ for each midpoint:
$f(3.25) = \frac{18}{3.25}, \quad f(3.75) = \frac{18}{3.75}, \quad f(4.25) = \frac{18}{4.25}, \quad f(4.75) = \frac{18}{4.75}$

STEP 6

Apply the Midpoint Rule to approximate the integral:
The Midpoint Rule formula is:
$\int_{a}^{b} f(x) \, dx \approx \Delta x \sum_{i=1}^{n} f(x_i^*)$
Substitute the values:
$\int_{3}^{5} \frac{18}{x} \, dx \approx 0.5 \left( \frac{18}{3.25} + \frac{18}{3.75} + \frac{18}{4.25} + \frac{18}{4.75} \right)$
Calculate each term:
$\approx 0.5 \left( 5.538 + 4.8 + 4.235 + 3.789 \right)$
Sum the terms:
$\approx 0.5 \times 18.362 = 9.181$
Round to three decimal places:
$\approx 9.181$

STEP 7

Verify the result using a graphing utility:
Use a graphing utility to compute $\int_{3}^{5} \frac{18}{x} \, dx$ and compare with the approximation.
The approximate value of the integral is:
$\boxed{9.181}$

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QuestionUse the Midpoint Rule with $n=4$ to approximate the value of the definite integral. Use a graphing utility to verify your result. (Round your answer to three decimal places.) $\int_{3}^{5} \frac{18}{x} d x$

Studdy Solution

STEP 1

1. We are asked to approximate the definite integral $\int_{3}^{5} \frac{18}{x} \, dx$ using the Midpoint Rule.
2. We will use $n = 4$ subintervals for the approximation.
3. We will round the final answer to three decimal places.
4. A graphing utility will be used to verify the result.

STEP 2

1. Determine the width of each subinterval.
2. Identify the midpoints of each subinterval.
3. Evaluate the function at each midpoint.
4. Apply the Midpoint Rule to approximate the integral.
5. Verify the result using a graphing utility.

STEP 3

Determine the width of each subinterval:
The interval $[3, 5]$ is divided into $n = 4$ subintervals. The width $\Delta x$ of each subinterval is given by:
$\Delta x = \frac{b-a}{n} = \frac{5-3}{4} = \frac{2}{4} = 0.5$

STEP 4

Identify the midpoints of each subinterval:
The subintervals are $[3, 3.5]$ , $[3.5, 4]$ , $[4, 4.5]$ , and $[4.5, 5]$ . The midpoints $x_i^$ are:
$x_1^ = 3.25, \quad x_2^* = 3.75, \quad x_3^* = 4.25, \quad x_4^* = 4.75$

STEP 5

Evaluate the function at each midpoint:
$f(x) = \frac{18}{x}$
Calculate $f(x_i^*)$ for each midpoint:
$f(3.25) = \frac{18}{3.25}, \quad f(3.75) = \frac{18}{3.75}, \quad f(4.25) = \frac{18}{4.25}, \quad f(4.75) = \frac{18}{4.75}$

STEP 6

STEP 7

Verify the result using a graphing utility:
Use a graphing utility to compute $\int_{3}^{5} \frac{18}{x} \, dx$ and compare with the approximation.
The approximate value of the integral is:
$\boxed{9.181}$

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QuestionUse the Midpoint Rule with n=4n=4n=4 to approximate the value of the definite integral. Use a graphing utility to verify your result. (Round your answer to three decimal places.) ∫3518xdx\int_{3}^{5} \frac{18}{x} d x∫35​x18​dx

Studdy Solution

STEP 1

STEP 2

1. Determine the width of each subinterval.2. Identify the midpoints of each subinterval.3. Evaluate the function at each midpoint.4. Apply the Midpoint Rule to approximate the integral.5. Verify the result using a graphing utility.

STEP 3

Determine the width of each subinterval:The interval [3,5][3, 5][3,5] is divided into n=4n = 4n=4 subintervals. The width Δx\Delta xΔx of each subinterval is given by:Δx=b−an=5−34=24=0.5\Delta x = \frac{b-a}{n} = \frac{5-3}{4} = \frac{2}{4} = 0.5Δx=nb−a​=45−3​=42​=0.5

STEP 4

STEP 5

STEP 6

STEP 7

Verify the result using a graphing utility:Use a graphing utility to compute ∫3518x dx\int_{3}^{5} \frac{18}{x} \, dx∫35​x18​dx and compare with the approximation. The approximate value of the integral is:9.181\boxed{9.181}9.181​

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QuestionUse the Midpoint Rule with $n=4$ to approximate the value of the definite integral. Use a graphing utility to verify your result. (Round your answer to three decimal places.) $\int_{3}^{5} \frac{18}{x} d x$

1. Determine the width of each subinterval.
2. Identify the midpoints of each subinterval.
3. Evaluate the function at each midpoint.
4. Apply the Midpoint Rule to approximate the integral.
5. Verify the result using a graphing utility.

Determine the width of each subinterval:
The interval $[3, 5]$ is divided into $n = 4$ subintervals. The width $\Delta x$ of each subinterval is given by:
$\Delta x = \frac{b-a}{n} = \frac{5-3}{4} = \frac{2}{4} = 0.5$

Verify the result using a graphing utility:
Use a graphing utility to compute $\int_{3}^{5} \frac{18}{x} \, dx$ and compare with the approximation.
The approximate value of the integral is:
$\boxed{9.181}$