Math / Calculus

QuestionExpress the limit as a definite integral on the given interval. $\begin{array}{c} \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[5\left(x_{i}^{}\right)^{3}-5 x_{i}^{}\right] \Delta x, \quad[2,6] \\ \left.\int_{2} \square\right) d x \end{array}$

Studdy Solution

STEP 1

1. We are given a limit of a Riemann sum.
2. The function inside the sum is $5(x_i^)^3 - 5x_i^$ .
3. The interval of integration is $[2, 6]$ .
4. We need to express the limit as a definite integral.

STEP 2

1. Identify the function $f(x)$ being integrated.
2. Identify the interval of integration.
3. Write the definite integral.

STEP 3

Identify the function $f(x)$ from the Riemann sum. The expression inside the sum is:
$f(x) = 5x^3 - 5x$

STEP 4

Identify the interval of integration from the problem statement, which is $[2, 6]$ .

STEP 5

Write the definite integral using the identified function and interval:
$\int_{2}^{6} (5x^3 - 5x) \, dx$
The limit of the Riemann sum as a definite integral is:
$\int_{2}^{6} (5x^3 - 5x) \, dx$

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QuestionExpress the limit as a definite integral on the given interval. $\begin{array}{c} \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[5\left(x_{i}^{}\right)^{3}-5 x_{i}^{}\right] \Delta x, \quad[2,6] \\ \left.\int_{2} \square\right) d x \end{array}$

Studdy Solution

STEP 1

1. We are given a limit of a Riemann sum.
2. The function inside the sum is $5(x_i^)^3 - 5x_i^$ .
3. The interval of integration is $[2, 6]$ .
4. We need to express the limit as a definite integral.

STEP 2

1. Identify the function $f(x)$ being integrated.
2. Identify the interval of integration.
3. Write the definite integral.

STEP 3

Identify the function $f(x)$ from the Riemann sum. The expression inside the sum is:
$f(x) = 5x^3 - 5x$

STEP 4

Identify the interval of integration from the problem statement, which is $[2, 6]$ .

STEP 5

Write the definite integral using the identified function and interval:
$\int_{2}^{6} (5x^3 - 5x) \, dx$
The limit of the Riemann sum as a definite integral is:
$\int_{2}^{6} (5x^3 - 5x) \, dx$

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Studdy Solution

STEP 1

1. We are given a limit of a Riemann sum.2. The function inside the sum is 5(xi∗)3−5xi∗ 5(x_i^*)^3 - 5x_i^* 5(xi∗​)3−5xi∗​.3. The interval of integration is [2,6][2, 6][2,6].4. We need to express the limit as a definite integral.

STEP 2

1. Identify the function f(x) f(x) f(x) being integrated.2. Identify the interval of integration.3. Write the definite integral.

STEP 3

Identify the function f(x) f(x) f(x) from the Riemann sum. The expression inside the sum is:f(x)=5x3−5x f(x) = 5x^3 - 5x f(x)=5x3−5x

STEP 4

Identify the interval of integration from the problem statement, which is [2,6][2, 6][2,6].

STEP 5

Write the definite integral using the identified function and interval:∫26(5x3−5x) dx \int_{2}^{6} (5x^3 - 5x) \, dx ∫26​(5x3−5x)dx The limit of the Riemann sum as a definite integral is:∫26(5x3−5x) dx \int_{2}^{6} (5x^3 - 5x) \, dx ∫26​(5x3−5x)dx

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1. We are given a limit of a Riemann sum.
2. The function inside the sum is $5(x_i^)^3 - 5x_i^$ .
3. The interval of integration is $[2, 6]$ .
4. We need to express the limit as a definite integral.

1. Identify the function $f(x)$ being integrated.
2. Identify the interval of integration.
3. Write the definite integral.

Identify the function $f(x)$ from the Riemann sum. The expression inside the sum is:
$f(x) = 5x^3 - 5x$

Identify the interval of integration from the problem statement, which is $[2, 6]$ .

Write the definite integral using the identified function and interval:
$\int_{2}^{6} (5x^3 - 5x) \, dx$
The limit of the Riemann sum as a definite integral is:
$\int_{2}^{6} (5x^3 - 5x) \, dx$