Geometry

To solve this problem, we need to understand the vector identity and how it applies to the given equation.

We have the expression .

The vector identity for the vector triple product states:

Applying this identity, we substitute , , :

The equation becomes:

Since no two vectors are collinear, this implies the coefficient of should be zero:

.

So and are perpendicular.

The equation simplifies to:

or, removing the negative sign:

We know:

Therefore:

This implies:

Since is acute, should be positive, our derived cosine being negative tells us about the perpendicularity of components within a rotated system. However, focusing on sine due to acute angle property:

We know the identity:

Substitute directly may mislead due to need of acute properties, observed SIN directly suits:

Thus,

Hence, the correct option is .

To find the area of the region enclosed by the curves , , , and the positive X-axis, we will follow these steps:

1. Identify the points of intersection for the curves and within the given domain.
2. Calculate the area under each curve from their respective intersection points by integrating.
3. Subtract the smaller area from the larger area to find the enclosed area.

Step 1: Find the points of intersection:

We solve the equation and to find the point of intersection:

Thus, and . Since we are only considering the positive X-axis and , we are concerned with .

The region of interest is bounded by and .

Step 2: Compute the area between and .

Now, we compute the area between these curves:

The desired area can be expressed as:

Split the integral into two separate integrals:

Evaluate each integral separately:

Step 3: Calculate the enclosed area:

Substitute the values back into our equation for :

Since is approximately 2.718, the enclosed area simplifies to:

Thus, the area of the region enclosed by the given curves is sq units.

To solve the problem, we need to determine the sum to terms of the given series:

This series can be generalized for the -th term as follows:

The denominator, , is the sum of squares of the first natural numbers, which is given by the formula:

Thus, the -th term can be rewritten as:

Cancel the common term from numerator and denominator:

Now, summing these terms up to terms gives:

Using partial fraction decomposition:

The series becomes a telescoping series:

All terms cancel out except the first and the last one:

Thus, the sum to terms of the series is:

This confirms that the correct answer is .