Math / Calculus

QuestionUse this definition with right endpoints to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit. $\begin{array}{l} f(x)=x \sqrt{x^{3}+5}, 1 \leq x \leq 4 \\ A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \square \end{array}$ Need Help? Read It

Studdy Solution

STEP 1

1. We are given a function $f(x) = x \sqrt{x^3 + 5}$ and an interval $[1, 4]$ .
2. We need to express the area under the curve as a limit of a Riemann sum using right endpoints.
3. We will not evaluate the limit, only express it.

STEP 2

1. Divide the interval $[1, 4]$ into $n$ subintervals.
2. Determine the width of each subinterval.
3. Identify the right endpoint of each subinterval.
4. Set up the Riemann sum using the right endpoints.
5. Express the area as a limit of the Riemann sum.

STEP 3

Divide the interval $[1, 4]$ into $n$ subintervals. The width of each subinterval, $\Delta x$ , is given by:
$\Delta x = \frac{4 - 1}{n} = \frac{3}{n}$

STEP 4

Determine the right endpoint of each subinterval. For the $i$ -th subinterval, the right endpoint $x_i$ is:
$x_i = 1 + i \Delta x = 1 + i \left(\frac{3}{n}\right)$

STEP 5

Set up the Riemann sum using the right endpoints. The Riemann sum $S_n$ is:
$S_n = \sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} \left( \left(1 + i \frac{3}{n}\right) \sqrt{\left(1 + i \frac{3}{n}\right)^3 + 5} \right) \frac{3}{n}$

STEP 6

Express the area $A$ as a limit of the Riemann sum:
$A = \lim_{n \to \infty} \sum_{i=1}^{n} \left( \left(1 + i \frac{3}{n}\right) \sqrt{\left(1 + i \frac{3}{n}\right)^3 + 5} \right) \frac{3}{n}$

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QuestionUse this definition with right endpoints to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit. $\begin{array}{l} f(x)=x \sqrt{x^{3}+5}, 1 \leq x \leq 4 \\ A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \square \end{array}$ Need Help? Read It

Studdy Solution

STEP 1

1. We are given a function $f(x) = x \sqrt{x^3 + 5}$ and an interval $[1, 4]$ .
2. We need to express the area under the curve as a limit of a Riemann sum using right endpoints.
3. We will not evaluate the limit, only express it.

STEP 2

1. Divide the interval $[1, 4]$ into $n$ subintervals.
2. Determine the width of each subinterval.
3. Identify the right endpoint of each subinterval.
4. Set up the Riemann sum using the right endpoints.
5. Express the area as a limit of the Riemann sum.

STEP 3

Divide the interval $[1, 4]$ into $n$ subintervals. The width of each subinterval, $\Delta x$ , is given by:
$\Delta x = \frac{4 - 1}{n} = \frac{3}{n}$

STEP 4

Determine the right endpoint of each subinterval. For the $i$ -th subinterval, the right endpoint $x_i$ is:
$x_i = 1 + i \Delta x = 1 + i \left(\frac{3}{n}\right)$

STEP 5

Set up the Riemann sum using the right endpoints. The Riemann sum $S_n$ is:
$S_n = \sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} \left( \left(1 + i \frac{3}{n}\right) \sqrt{\left(1 + i \frac{3}{n}\right)^3 + 5} \right) \frac{3}{n}$

STEP 6

Express the area $A$ as a limit of the Riemann sum:
$A = \lim_{n \to \infty} \sum_{i=1}^{n} \left( \left(1 + i \frac{3}{n}\right) \sqrt{\left(1 + i \frac{3}{n}\right)^3 + 5} \right) \frac{3}{n}$

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Studdy Solution

STEP 1

1. We are given a function f(x)=xx3+5 f(x) = x \sqrt{x^3 + 5} f(x)=xx3+5​ and an interval [1,4][1, 4][1,4].2. We need to express the area under the curve as a limit of a Riemann sum using right endpoints.3. We will not evaluate the limit, only express it.

STEP 2

1. Divide the interval [1,4][1, 4][1,4] into nnn subintervals.2. Determine the width of each subinterval.3. Identify the right endpoint of each subinterval.4. Set up the Riemann sum using the right endpoints.5. Express the area as a limit of the Riemann sum.

STEP 3

Divide the interval [1,4][1, 4][1,4] into nnn subintervals. The width of each subinterval, Δx\Delta xΔx, is given by:Δx=4−1n=3n \Delta x = \frac{4 - 1}{n} = \frac{3}{n} Δx=n4−1​=n3​

STEP 4

Determine the right endpoint of each subinterval. For the iii-th subinterval, the right endpoint xix_ixi​ is:xi=1+iΔx=1+i(3n) x_i = 1 + i \Delta x = 1 + i \left(\frac{3}{n}\right) xi​=1+iΔx=1+i(n3​)

STEP 5

STEP 6

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1. We are given a function $f(x) = x \sqrt{x^3 + 5}$ and an interval $[1, 4]$ .
2. We need to express the area under the curve as a limit of a Riemann sum using right endpoints.
3. We will not evaluate the limit, only express it.

1. Divide the interval $[1, 4]$ into $n$ subintervals.
2. Determine the width of each subinterval.
3. Identify the right endpoint of each subinterval.
4. Set up the Riemann sum using the right endpoints.
5. Express the area as a limit of the Riemann sum.

Divide the interval $[1, 4]$ into $n$ subintervals. The width of each subinterval, $\Delta x$ , is given by:
$\Delta x = \frac{4 - 1}{n} = \frac{3}{n}$

Determine the right endpoint of each subinterval. For the $i$ -th subinterval, the right endpoint $x_i$ is:
$x_i = 1 + i \Delta x = 1 + i \left(\frac{3}{n}\right)$