Area Between Two Curves

We need to find the area of the region bounded by the line , the x-axis, and the ordinates and .

First, find where the line intersects the x-axis (y = 0):

So the line crosses the x-axis at .

This means from x = -1 to x = -1/2, the line is below the x-axis, and from x = -1/2 to x = 1, the line is above the x-axis.

The area is given by the sum of the absolute values of the integrals from -1 to -1/2 and from -1/2 to 1:

Area =

The antiderivative of is .

So the absolute value of the first integral is .

The total area is

Therefore, the correct option is (C) .

We need to find the area of the region bounded by the curve and the line .

First, find the points of intersection by setting the equations equal to each other:

and . Substitute into the first equation:

or

When , .

When , .

So, the points of intersection are (0, 0) and (2, 4).

Now we set up the integral for the area between the curves. Since , .

We want to integrate with respect to y, so we express x in terms of y. The line is .

We integrate from y = 0 to y = 4. The area is given by:

Therefore, the area of the region is square units.

Thus, the correct option is (B) sq. units.

The correct answer is: (C): log 2 sq. units.

We are tasked with finding the area between the curves and in the interval , and the X-axis.

Step 1: Set up the integral for the area between the curves

The area between two curves is given by the integral of the difference between the two functions. We need to integrate the difference between and over the interval . The area is therefore:

Step 2: Break the integral into two parts

We can separate the integral into two parts: one for and one for :

Step 3: Compute the integrals

We know that the integral of is and the integral of is . So we have:

Step 4: Evaluate the integrals

Now, substitute the limits and into both integrals:

Evaluating each integral at the limits gives:

At , and , and at , and . Substituting these values gives:

Thus, we find that the area is equal to:

Conclusion:
The area between the curves and in the interval is square units, so the correct answer is (C): log 2 sq. units.

The correct answer is: (B): 10 sq. units.

We are tasked with finding the area of the region bounded by the line and the vertical lines and .

Step 1: Set up the integral

We know that the area under the curve between two vertical lines is given by the integral of the function between the limits of the lines. The function is , and the limits are and . Therefore, we set up the integral for the area as follows:

Step 2: Compute the integral

Now, we compute the integral:

Step 3: Evaluate the integral

Substitute the limits and into the integral:

Evaluating at and :

Conclusion:
The area of the region is square units, so the correct answer is (B): 10 sq. units.