Find The Area Of The Region Enclosed By The Curve Y = Cos X

Find The Area Of The Region Enclosed By The Curve Yx Cos X And T

Find the area of the region enclosed by the curve y = x cos x and the x-axis for π ≤ x ≤ π.

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The problem requires calculating the area enclosed by the curve y = x cos x and the x-axis over the interval π ≤ x ≤ π. To solve this, we need to understand where the function y = x cos x intersects the x-axis within the given interval, set up the appropriate definite integral, and evaluate it accordingly.

First, let's analyze the behavior of y = x cos x. The roots of this function, i.e., the points where y = 0, occur when either x = 0 or cos x = 0. Since the interval is from π to π, and both π and π are the same point, we interpret the problem as potentially from -π to π or considering symmetric properties; however, given the initial phrasing, assume the interval is from -π to π for a meaningful enclosed area calculation. If the interval is from -π to π, then we consider the zeros of y = x cos x within this domain.

The zeros of y = x cos x occur at x = 0 and where cos x = 0, which are at x= ±π/2, ±3π/2, etc. Within -π to π, cos x = 0 at x= ±π/2. Since at these points the function y = x cos x crosses the x-axis, the enclosed regions are between these zeros.

The critical points are therefore at x = -π/2, 0, and π/2. On the interval from -π to π, the function y = x cos x changes sign at these zeros, and the absolute value of the integral over each subinterval will give the area segments. The total enclosed area is obtained by summing the absolute values of the integrals over the subintervals where the function is positive or negative.

Set up the integral as:

Area = ∫_{-π}^{-π/2} |x cos x| dx + ∫_{-π/2}^{0} |x cos x| dx + ∫_{0}^{π/2} |x cos x| dx + ∫_{π/2}^{π} |x cos x| dx

Since the function mays change signs at these points, we evaluate the integrals with proper signs, or equivalently, take absolute values after computing the integrals. Due to symmetry, the areas over symmetric intervals might simplify calculations.

Alternatively, recognizing the symmetry of the sine and cosine functions and the function y = x cos x, which is an odd function (since x is odd, and cos x is even, their product is odd), the entire integral from -π to π of |x cos x| dx is twice the integral from 0 to π, considering the absolute value.

Indeed, because y = x cos x is odd, its integral from -a to a is zero, but the area (the absolute value integral) from -a to a is twice the integral from 0 to a of |x cos x| dx.

Therefore, the area is:

Area = 2 ∫_{0}^{π} |x cos x| dx

Next, determine the sign of x cos x on 0 to π:

  • In 0 0, so x cos x > 0, and |x cos x| = x cos x.
  • In π/2

Thus, the area is:

Area = 2 [∫_{0}^{π/2} x cos x dx + ∫_{π/2}^{π} - x cos x dx]

Now, evaluate these integrals.

Calculating ∫ x cos x dx

Use integration by parts:

Let u = x, dv = cos x dx ⇒ du = dx, v = sin x.

Then:

∫ x cos x dx = x sin x - ∫ sin x dx = x sin x + cos x + C

Next, evaluate the definite integrals:

First integral:

∫_{0}^{π/2} x cos x dx = [x sin x + cos x]_{0}^{π/2}

At x = π/2:

π/2 * 1 + 0 = π/2

At x = 0:

0 * 0 + 1 = 1

Thus, ∫_{0}^{π/2} x cos x dx = π/2 - 1

Second integral:

∫_{π/2}^{π} - x cos x dx = - [x sin x + cos x]_{π/2}^{π}

At x = π:

π * 0 + (-1) = -1

At x = π/2:

π/2 * 1 + 0 = π/2

Difference:

(-1) - (π/2) = -1 - π/2

Multiplying by -1 (due to the negative sign outside the integral):

= 1 + π/2

Now, the total enclosed area becomes:

Area = 2 [ (π/2 - 1) + (1 + π/2) ] = 2 [ (π/2 - 1 + 1 + π/2) ] = 2 [ (π/2 + π/2) ] = 2 [ π ] = 2π

Hence, the area enclosed by the curve y = x cos x and the x-axis over the interval -π to π is 2π.

Conclusion

The key steps involved analyzing the symmetry of the function, identifying the points where it crosses the x-axis, setting up the integrals over the appropriate subintervals, and employing integration by parts to evaluate the definite integrals accurately. The result is that the total area enclosed by the curve and the x-axis over the specified interval is 2π square units.

References

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