Question

Find The Area Of The Region Bounded By The X-Axis And The Curves Y = 4 Sin(X) And Y = 4 Cos(X) Where X Element [0, Pi/2. A) 4 - 2 Squareroot 2 B) 16 - 8 Squareroot 2 C) 2 Squareroot 2 D) 8 - 4 Squareroot 2 E) 8

Answer 1

To find the area of the region bounded by the x-axis and the curves y = 4 sin(x) and y = 4 cos(x), we need to determine the points of intersection and integrate the difference between the two curves with respect to x. The area is equal to 16 - 8√2.

Step-by-step explanation:

To find the area of the region bounded by the x-axis and the curves y = 4 sin(x) and y = 4 cos(x), we need to determine the points of intersection of these curves. Since the given range is [0, Pi/2], we can see that the curves intersect at x = 0 and x = Pi/4. By integrating the difference between the two curves with respect to x from 0 to Pi/4, we can calculate the area.

Let's start by finding the points of intersection:

y = 4 sin(x) and y = 4 cos(x)

4 sin(x) = 4 cos(x)

Simplifying, we get sin(x) = cos(x)

Using the identity sin(x) = cos(π/2 - x), we have cos(π/2 - x) = cos(x)

Therefore, π/2 - x = x

2x = π/2

x = π/4

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

Next, we integrate the difference between the two curves with respect to x:

Area = ∫[0, π/4] (4 cos(x) - 4 sin(x)) dx

Using integration rules, we get [4 sin(x) + 4 cos(x)] evaluated from 0 to π/4

Area = (4 sin(π/4) + 4 cos(π/4)) - (4 sin(0) + 4 cos(0))

Area = (4/√2 + 4/√2) - (0 + 4)

Area = 8/√2 - 4

Area = 4√2 - 4

Hence, the answer is option B) 16 - 8√2.