Math / Calculus

Question\begin{tabular}{|l|l|} \hline & \\ \hline 1 & $\int_{0}^{3} x^{3} d x$ \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. We are given a definite integral problem.
2. The function to be integrated is $x^3$ .
3. The limits of integration are from 0 to 3.

STEP 2

1. Set up the integral.
2. Find the antiderivative of the function.
3. Evaluate the antiderivative at the upper and lower limits.
4. Subtract the values to find the definite integral.

STEP 3

Set up the integral with the given limits and function:
$\int_{0}^{3} x^{3} \, dx$

STEP 4

Find the antiderivative of $x^3$ . The antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1} + C$ , where $C$ is the constant of integration. For $x^3$ , the antiderivative is:
$\frac{x^{4}}{4} + C$

STEP 5

Evaluate the antiderivative at the upper limit (3) and the lower limit (0):
At $x = 3$ : $\frac{3^{4}}{4} = \frac{81}{4}$
At $x = 0$ : $\frac{0^{4}}{4} = 0$

STEP 6

Subtract the value of the antiderivative at the lower limit from the value at the upper limit:
$\frac{81}{4} - 0 = \frac{81}{4}$
The value of the definite integral is:
$\boxed{\frac{81}{4}}$

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Question\begin{tabular}{|l|l|} \hline & \\ \hline 1 & $\int_{0}^{3} x^{3} d x$ \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. We are given a definite integral problem.
2. The function to be integrated is $x^3$ .
3. The limits of integration are from 0 to 3.

STEP 2

1. Set up the integral.
2. Find the antiderivative of the function.
3. Evaluate the antiderivative at the upper and lower limits.
4. Subtract the values to find the definite integral.

STEP 3

Set up the integral with the given limits and function:
$\int_{0}^{3} x^{3} \, dx$

STEP 4

Find the antiderivative of $x^3$ . The antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1} + C$ , where $C$ is the constant of integration. For $x^3$ , the antiderivative is:
$\frac{x^{4}}{4} + C$

STEP 5

Evaluate the antiderivative at the upper limit (3) and the lower limit (0):
At $x = 3$ : $\frac{3^{4}}{4} = \frac{81}{4}$
At $x = 0$ : $\frac{0^{4}}{4} = 0$

STEP 6

Subtract the value of the antiderivative at the lower limit from the value at the upper limit:
$\frac{81}{4} - 0 = \frac{81}{4}$
The value of the definite integral is:
$\boxed{\frac{81}{4}}$

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Question\begin{tabular}{|l|l|} \hline & \\ \hline 1 & ∫03x3dx\int_{0}^{3} x^{3} d x∫03​x3dx \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. We are given a definite integral problem.2. The function to be integrated is x3 x^3 x3.3. The limits of integration are from 0 to 3.

STEP 2

1. Set up the integral.2. Find the antiderivative of the function.3. Evaluate the antiderivative at the upper and lower limits.4. Subtract the values to find the definite integral.

STEP 3

Set up the integral with the given limits and function:∫03x3 dx \int_{0}^{3} x^{3} \, dx ∫03​x3dx

STEP 4

Find the antiderivative of x3 x^3 x3. The antiderivative of xn x^n xn is xn+1n+1+C \frac{x^{n+1}}{n+1} + C n+1xn+1​+C, where C C C is the constant of integration. For x3 x^3 x3, the antiderivative is:x44+C \frac{x^{4}}{4} + C 4x4​+C

STEP 5

Evaluate the antiderivative at the upper limit (3) and the lower limit (0):At x=3 x = 3 x=3: 344=814 \frac{3^{4}}{4} = \frac{81}{4} 434​=481​At x=0 x = 0 x=0: 044=0 \frac{0^{4}}{4} = 0 404​=0

STEP 6

Subtract the value of the antiderivative at the lower limit from the value at the upper limit:814−0=814 \frac{81}{4} - 0 = \frac{81}{4} 481​−0=481​ The value of the definite integral is:814 \boxed{\frac{81}{4}} 481​​

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Question\begin{tabular}{|l|l|} \hline & \\ \hline 1 & $\int_{0}^{3} x^{3} d x$ \\ \hline \end{tabular}

1. We are given a definite integral problem.
2. The function to be integrated is $x^3$ .
3. The limits of integration are from 0 to 3.

1. Set up the integral.
2. Find the antiderivative of the function.
3. Evaluate the antiderivative at the upper and lower limits.
4. Subtract the values to find the definite integral.

Set up the integral with the given limits and function:
$\int_{0}^{3} x^{3} \, dx$

Find the antiderivative of $x^3$ . The antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1} + C$ , where $C$ is the constant of integration. For $x^3$ , the antiderivative is:
$\frac{x^{4}}{4} + C$

Evaluate the antiderivative at the upper limit (3) and the lower limit (0):
At $x = 3$ : $\frac{3^{4}}{4} = \frac{81}{4}$
At $x = 0$ : $\frac{0^{4}}{4} = 0$

Subtract the value of the antiderivative at the lower limit from the value at the upper limit:
$\frac{81}{4} - 0 = \frac{81}{4}$
The value of the definite integral is:
$\boxed{\frac{81}{4}}$