Question

Evaluate the integral: \[ \int_0^1 \int_x^1 \sin(y^2) \, dy \, dx \]

• A. \( \frac{1 + \cos 1}{2} \)
• B. \( 1 - \cos 1 \)
• C. \( 1 + \cos 1 \)
• D. \( \frac{1 - \cos 1}{2} \)

Hint:

When integrating over two variables, sometimes changing the order of integration simplifies the problem.

Answer 1

The Correct Option is D

Step 1: Changing the order of integration.
The integral can be rewritten by changing the order of integration. The limits of integration suggest that \( y \) goes from \( x \) to 1, and \( x \) goes from 0 to 1. Changing the order: \[ \int_0^1 \int_x^1 \sin(y^2) \, dy \, dx = \int_0^1 \int_0^y \sin(y^2) \, dx \, dy. \] Now, the inner integral with respect to \( x \) is straightforward: \[ \int_0^y 1 \, dx = y. \] Thus, the integral simplifies to: \[ \int_0^1 y \sin(y^2) \, dy. \]
Step 2: Substituting and solving the integral.
Let \( u = y^2 \), so \( du = 2y \, dy \). The limits for \( u \) are from 0 to 1. Thus, the integral becomes: \[ \frac{1}{2} \int_0^1 \sin(u) \, du. \] This integrates to: \[ \frac{1}{2} [-\cos(u)]_0^1 = \frac{1}{2} (1 - \cos(1)). \]
Step 3: Conclusion.
Thus, the correct answer is (D) \( \frac{1 - \cos 1}{2} \).