Question

The value of \(\iint_R x^2 y^5 \, dx \, dy\), where R is the rectangle with vertices (1, 0), (1, 1), (2, 1) and (2, 0) is:

• A. 1
• B. 2
• C. 7/18
• D. 5/18

Hint:

Exam Tip:
For double integrals over rectangles:

• The limits of integration are constant.
• The order of integration can be swapped.
• Integrate one variable at a time.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to evaluate a double integral over a rectangular region.

Step 2: Key Formula or Approach:
The region R is the rectangle: \(1 \le x \le 2\), \(0 \le y \le 1\).
The integral is: \[ \int_{x=1}^{2} \int_{y=0}^{1} x^2 y^5 \, dy \, dx \]

Step 3: Detailed Explanation:
First, integrate with respect to \(y\): \[ \int_0^1 y^5 \, dy = \left[ \frac{y^6}{6} \right]_0^1 = \frac{1}{6} \] Now, integrate with respect to \(x\): \[ \int_1^2 x^2 \cdot \frac{1}{6} \, dx = \frac{1}{6} \left[ \frac{x^3}{3} \right]_1^2 = \frac{1}{18} (8 - 1) = \frac{7}{18} \]

Step 4: Final Answer:
Therefore, option (C) is correct.