Find the area enclosed by the curve y=sin x and the x-axis between the x=0 and the x=pie.

To find the area enclosed by the curve \( y = \sin x \) and the x-axis from \( x = 0 \) to \( x = \pi \), we need to set up an integral. The area under the curve can be calculated using the definite integral of the function over the specified interval.

Setting Up the Integral

The area \( A \) can be expressed mathematically as:

A = ∫ from 0 to π of sin(x) dx

Calculating the Integral

To compute this integral, we first need to find the antiderivative of \( \sin x \). The antiderivative of \( \sin x \) is \( -\cos x \). Therefore, we can write:

A = [-cos(x)] from 0 to π

Evaluating the Antiderivative

Now, we will evaluate this expression at the bounds of the integral:

• At \( x = \pi \): \( -\cos(\pi) = -(-1) = 1 \)
• At \( x = 0 \): \( -\cos(0) = -1 \)

Putting these values into our expression gives us:

A = 1 - (-1) = 1 + 1 = 2

Final Result

The area enclosed by the curve \( y = \sin x \) and the x-axis between \( x = 0 \) and \( x = \pi \) is equal to 2 square units. This area represents the region above the x-axis, where the sine function is positive in this interval.

Visualizing the Area

To better understand this, you can visualize the sine curve starting at the origin (0,0), peaking at (π/2, 1), and returning to the x-axis at (π,0). The area under this curve forms a wave-like shape, and the integral effectively sums up all the infinitesimally small rectangles under the curve from 0 to π, resulting in the total area of 2.