Question

The value of \(\int_{1}^{3} x^3 \, dx\) will be

• A. 80
• B. 20
• C. 27
• D. 104

Hint:

Double-check simple powers to save time:
\(3^4 = 81\) and \(1^4 = 1\).
The difference is \(80\), which is immediately divisible by \(4\) to give \(20\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The definite integral computes the net signed area under the curve described by a function within the specified interval limits.

Step 2: Key Formula or Approach:
The power rule of integration is:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (\text{for } n \neq -1) \] For a definite integral:
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]

Step 3: Detailed Explanation:
Let us perform the integration of the given function:
\[ \int_{1}^{3} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{1}^{3} \] Now, substitute the upper limit \(3\) and lower limit \(1\):
\[ \left[ \frac{x^4}{4} \right]_{1}^{3} = \frac{3^4}{4} - \frac{1^4}{4} \] \[ = \frac{81}{4} - \frac{1}{4} \] \[ = \frac{80}{4} = 20 \]

Step 4: Final Answer:
The correct option is 2, which corresponds to 20.