𝐅𝐢𝐧𝐝 𝐭𝐡𝐞 𝐚𝐫𝐞𝐚 𝐨𝐟 𝐭𝐡𝐞 𝐟𝐢𝐧𝐢𝐭𝐞 𝐫𝐞𝐠𝐢𝐨𝐧 𝐛𝐨𝐮𝐧𝐝𝐞𝐝 𝐛𝐲 𝐭𝐡𝐞 𝐜𝐮𝐫𝐯𝐞 𝐚𝐧𝐝 𝐭𝐡𝐞 𝐗 𝐚𝐱𝐢𝐬 𝐘= 8𝐱-𝐱²-12

To find the area of the finite region bounded by the curve Y = 8x - x^2 - 12 and the x-axis, we first need to find the x-intercepts of the curve. These will be our limits of integration. Step 1: Find the x-intercepts. Set Y = 0: 8x - x^2 - 12 = 0 Rearrange the terms into standard quadratic form: -x^2 + 8x - 12 = 0 Multiply by -1 to make the leading coefficient positive: x^2 - 8x + 12 = 0 Factor the quadratic equation: (x - 2)(x - 6) = 0 This gives us the x-intercepts: x = 2 and x = 6 These are the limits of integration. Step 2: Set up the definite integral. The area A is given by the integral of the function from x=2 to x=6: A = _2^6 (8x - x^2 - 12)\,dx Step 3: Evaluate the integral. First, find the antiderivative of the function: (8x - x^2 - 12)\,dx = 8(x^2)/(2) - (x^3)/(3) - 12x + C = 4x^2 - (x^3)/(3) - 12x + C Now, evaluate the definite integral using the limits: A = [ 4x^2 - (x^3)/(3) - 12x ]_2^6 Substitute the upper limit (x=6): F(6) = 4(6)^2 - ((6)^3)/(3) - 12(6) F(6) = 4(36) - (216)/(3) - 72 F(6) = 144 - 72 - 72 F(6) = 0 Substitute the lower limit (x=2): F(2) = 4(2)^2 - ((2)^3)/(3) - 12(2) F(2) = 4(4) - (8)/(3) - 24 F(2) = 16 - (8)/(3) - 24 F(2) = -8 - (8)/(3) F(2) = -(24)/(3) - (8)/(3) F(2) = -(32)/(3) Finally, subtract F(2) from F(6): A = F(6) - F(2) A = 0 - (-(32)/(3)) A = (32)/(3) The area of the finite region is (32)/(3). Send me the next one 📸