Applications Of Integrals

The region is bounded by the y-axis on one side, so we integrate with respect to x.
The curve y = cos(x) is above the curve y = sin(x) in the given interval.
Let's denote the area as A. To calculate it, we can set up the following integral:
A = ∫[0, ] (cos(x) - sin(x)) dx
To evaluate this integral, we take the antiderivative of cos(x) and sin(x):
A = [sin(x) + cos(x)]|[0, ]
Now we substitute the limits of integration:
A = [sin + cos ] - [sin(0) + cos(0)] = [ + ] - [0 + 1] = - 1
Therefore, the area of the region bounded by the y-axis, y = cos(x), and y = sin(x) for 0 ≤ x ≤ π/4 is ( - 1) square units.
Hence, the correct answer is option (C) ( - 1) sq. units.

The area of triangle ABC, bounded by the lines , , and , can be calculated as follows:

Step 1: Understanding the Problem
The area of triangle ABC can be determined by first calculating the area of the regions formed by the lines and the x-axis, then subtracting the area of one triangle from another.

Step 2: Define the areas of triangles ABED and ACED
The area of triangle ABED is the integral of the function minus 0 with respect to , evaluated from to . Similarly, the area of triangle ACED is the integral of the function minus 0 with respect to , also evaluated from to .

Step 3: Set up the integrals
The area of triangle ABED is: The area of triangle ACED is:

Step 4: Simplify the integrals
Expanding the integrals:

Step 5: Evaluate the integrals
First, evaluate the integral for : So: Next, evaluate the integral for : So:

Step 6: Find the area of triangle ABC
The area of triangle ABC is the difference between the areas of triangle ABED and triangle ACED:

Final Answer:
Therefore, the area of triangle ABC is square units.