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Step-5

Title: Integration

Grade: 1300-a Lesson: S2-L6

Explanation: Hello Students, time to practice and review the steps for the problem.

Lesson Steps

Step	Type	Explanation	Answer
1	Problem	Evaluate \$ \int (x^2) dx\$ with limits from 1 to 3.
2	Formula:	The power rule	\$\int x^n = x^(n+1)/(n+1) + C \$
3	Hint	Now apply the power rule after simplification to find the antiderivative of \$x^2\$	\$ \int (x^2) dx = x^(2+1)/(2+1) + C\$ \$ \int (x^2) dx = x^3/3 + C\$
4	Step	To evaluate the definite integral from 1 to 3, we substitute limits, apply antiderivative, and find the difference.	\$ 1/3(x^3)_1^3 = 1/3 (3^3 - 1^3)\$ \$ 1/3(x^3)_1^3 = 1/3 (27 - 1)\$
5	Step	After simplification	\$ 1/3(x^3)_1^3 = 1/3 (26)\$ \$ 1/3(x^3)_1^3 = 26/3 \$
6	Step	Therefore, the value of \$ \int (x^2) dx\$ with limits from 1 to 3 is \$26/3 \$.
7	Choice.A	This statement is inaccurate due to an error in the value, it needs correction from \$23/3\$ to \$26/3\$ for accuracy	\$23/3\$
8	Choice.B	This statement is inaccurate due to an error in the value, it needs correction from \$28/3\$ to \$26/3\$ for accuracy	\$28/3\$
9	Choice.C	When you plug in the specified values, the definite integral yields a result of \$26/3\$	\$26/3\$
10	Choice.D	This statement is inaccurate due to an error in the value, it needs correction from \$35/3\$ to \$26/3\$ for accuracy	\$35/3\$
11	Answer	Option	C
12	Sumup	Can you summarize what you’ve understood in the above steps?

Step

Type

Explanation

Answer

Problem

Evaluate \$ \int (x^2) dx\$ with limits from 1 to 3.

Formula:

The power rule

\$\int x^n = x^(n+1)/(n+1) + C \$

Hint

Now apply the power rule after simplification to find the antiderivative of \$x^2\$

\$ \int (x^2) dx = x^(2+1)/(2+1) + C\$

\$ \int (x^2) dx = x^3/3 + C\$

Step

To evaluate the definite integral from 1 to 3, we substitute limits, apply antiderivative, and find the difference.

\$ 1/3(x^3)_1^3 = 1/3 (3^3 - 1^3)\$

\$ 1/3(x^3)_1^3 = 1/3 (27 - 1)\$

Step

After simplification

\$ 1/3(x^3)_1^3 = 1/3 (26)\$

\$ 1/3(x^3)_1^3 = 26/3 \$

Step

Therefore, the value of \$ \int (x^2) dx\$ with limits from 1 to 3 is \$26/3 \$.

Choice.A

This statement is inaccurate due to an error in the value, it needs correction from \$23/3\$ to \$26/3\$ for accuracy

\$23/3\$

Choice.B

This statement is inaccurate due to an error in the value, it needs correction from \$28/3\$ to \$26/3\$ for accuracy

\$28/3\$

Choice.C

When you plug in the specified values, the definite integral yields a result of \$26/3\$

\$26/3\$

Choice.D

This statement is inaccurate due to an error in the value, it needs correction from \$35/3\$ to \$26/3\$ for accuracy

\$35/3\$

Answer

Option

Sumup

Can you summarize what you’ve understood in the above steps?