Question

One of the numbers \(\pi, \pi / 2,35 \pi / 128,1-\pi\) is the correct value of the integral \(\int_{0}^{\pi} \sin ^{8} x d x\). Use the graph of \(\mathrm{y}=\sin ^{8} \mathrm{x}\) and a logical process of elimination to find the correct value.

Short answer

A) \(\pi\) B) \(\pi/2\) C) \(35\pi/128\) D) \(1-\pi\) Answer: C) \(35\pi/128\)

Step-by-step solution

Graphing the function y = sin^8(x)

To get a better understanding of the function, let's first graph y = sin^8(x) on the domain [0, π]. You may use graphing software or a graphing calculator to plot the function. Once plotted, you'll notice that the function: - Has values in the range [0, 1] on this domain since sin^8(x) is always non-negative. - Is symmetric about the vertical line x = π/2. - Peaks at x = π/2 where sin^8(π/2) = 1.

Observing the properties of sin^8(x)

The graph of y = sin^8(x) has unique properties that will help us estimate the value of the integral. 1. Since sin^8(x) is always non-negative on [0, π], the integral will also be non-negative. So, we can immediately rule out any negative answers. 2. The graph is symmetric about x = π/2, meaning there's equal area on both sides of the line x = π/2. So the integral must be smaller than the area of a rectangle of length π (the interval of integration) and width 1 (peak value of the function). This area is π, so the integral must be smaller than π.

Process of elimination

With these observations, let's analyze each given possible value and compare them to the properties of sin^8(x): 1. \(\pi\): Potential value, as it is positive and less than π. 2. \(\pi/2\): Potential value, as it is positive and less than π. 3. \(35\pi/128\): Potential value, as it is positive and less than π. 4. \(1-\pi\): This value is negative, which contradicts the property of sin^8(x) being non-negative; so, it cannot be the correct integral value. Now, let's compare these potential values to the area of the rectangle mentioned in Step 2. We know that the integral must be smaller than π, but how much smaller? Looking back at the graph, we see that the graph of sin^8(x) closely hugs the x-axis and the left and right sides near 0 and π, so the integral must be much smaller than the area of the rectangle with the area of π. Comparing the potential values: 1. \(\pi\): Almost equal to the area of the rectangle. 2. \(\pi/2\): Still quite large compared to the area under the curve. 3. \(35\pi/128\): Significantly smaller than π; roughly 27.34% of its value. This value appears to be a more reasonable estimate of the integral. After considering the graph and using the process of elimination, we can conclude that the correct value of the integral is \(35\pi/128\).