Calculus

Step 1: Understanding the integral.
The vector field is given in cylindrical coordinates. The integration involves calculating the volume integral of the vector field over the cylindrical volume. Since is an odd function and is being integrated over the entire angular range from to , the integral will result in zero.
Step 2: Conclusion.
Thus, the integral evaluates to zero, and the correct answer is option (B).

Step 1: Set up the integral.
We are given the function and the region of integration . The volume integral is: Step 2: Perform the integration over .
The first part of the integral involves the terms: Step 3: Perform the integration over .
Now integrate with respect to : This results in: Step 4: Perform the integration over .
Now integrate over : Step 5: Final Calculation.
Now, combining all parts: Step 6: Conclusion.
Thus, the value of the integral is .

Step 1: Expand the inner function.
We use the Taylor series of . Thus, becomes .

Step 2: Expand the outer sine function.
, where . Substitute to get:
.

Step 3: Collect the terms.
First term gives .
Second term gives .

Step 4: Add the contributions.
Coefficient = .
Magnitude = 0.17 (up to two decimal points as required).

Step 1: Use general polynomial identity.
For , there is no term. Thus the sum of all roots = 0 (true).

Step 2: Product of roots.
Product = → matches (B), but note that sign is correct only if all roots counted (yes). So (B) is also correct mathematically.

Step 3: Complex-root property.
Real coefficients → complex roots occur in conjugate pairs ⇒ (D) true.

Step 4: Conclusion.
(A) and (D) are always true from polynomial structure.