Area Under Simple Curves

The given curve is . Since must be non-negative ( ), this equation represents the upper part of a curve.

To identify the shape of the curve, we can square both sides of the equation: Rearranging the terms, we get:

This is the equation of a circle centered at the origin (0, 0) with radius such that . Therefore, the radius is .

The equation represents the upper semi-circle ( ) of this circle. The region bounded by and the x-axis ( ) is the area of this upper semi-circle.

The area of a full circle with radius is given by the formula . For , the area of the full circle is square units.

The area of the region required (the upper semi-circle) is half the area of the full circle: square units.

The calculated area of the region bounded by and the x-axis is square units.

The equation can be rewritten as:

Comparing this with the standard form
we have

To find the integrating factor (IF), we calculate the exponential of the integral of

Multiplying both sides of the differential equation by the integrating factor, we get:

Now, the left side of the equation is the derivative of the product with respect to x:

Integrating both sides, we have:

Using integration by parts, we can evaluate the integral on the right side:

=

=

=

To find the value of C, we can use the initial condition

=

=

=
=

Therefore, the solution to the differential equation is:

To find y(e), we substitute

=

=

So,
Therefore, the value of y(e) is approximately 2.35067, which corresponds to option (B) 2.

The region lies between and .

We can find the area by integrating the absolute value of the function within this interval.

Thus, the area (A) is given by:

To evaluate this integral, we need to break it up into two parts due to the nature of the tangent function.

Simplifying this expression, we have:

Combining the integrals, we get:

Using the integral property, we have:

Therefore, the area of the region bounded by the curve , the x-axis, and the line is , which corresponds to option (C) .