Questions: Question Let R be the region bounded below by the graph of the function (f(x)=-fracx^26-2) and above by the (x)-axis over the interval ([-3,3]). Find the centroid, ((barx, bary)), of the region. Enter answer using exact values. Provide your answer below: ((barx, bary)=).

Solution Steps

The area of the region R is given by the definite integral of f(x) from -3 to 3. Since the region is bounded by the x-axis from above, we take the negative of the function because the region lies below the x-axis.

A = ∫_-3³ -f(x) dx = ∫_-3³ (-(-x²/6 - 2))dx = ∫_-3³ (x²/6 + 2) dx = [x³/18 + 2x]_-3³ = (27/18 + 6) - (-27/18 - 6) = (3/2 + 6) - (-3/2 - 6) = 15

Since the function f(x) is even and the interval [-3, 3] is symmetric around x=0, the x coordinate of the centroid is 0. This can also be seen from the graph as the region is symmetric with respect to the y-axis.

x̄ = (1/A) * ∫_-3³ x * (-f(x)) dx = 0

ȳ = (1/A) * (1/2) * ∫_-3³ (-f(x))²dx = (1/15) * (1/2) ∫_-3³ (x²/6 + 2)² dx ȳ = (1/30)∫_-3³ (x⁴/36 + (2/3)x² + 4)dx ȳ = (1/30)[x⁵/180 + (2/9)x³ + 4x]_-3³ ȳ = (1/30)[(243/180 + 54/9 + 12) - (-243/180 - 54/9 - 12)] ȳ = (1/30)(243/90 + 12 + 24) ȳ = (1/30)(27/10 + 36) ȳ = (27/300 + 36/30) ȳ = (27 + 360)/300 = 387/300 = 129/100 = -1.29

Final Answer: