This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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8) Step 1: Set up the definite integral for the area under the curve y = x^2 + 7 from x=1 to x=2. A = _1^2 (x^2 + 7)\,dx Step 2: Integrate the function term by term. (x^2 + 7)\,dx = x^2+12+1 + 7x = (x^3)/(3) + 7x Step 3: Evaluate the definite integral using the Fundamental Theorem of Calculus. A = [ (x^3)/(3) + 7x ]_1^2 = ( (2^3)/(3) + 7(2) ) - ( (1^3)/(3) + 7(1) ) Step 4: Calculate the values. A = ( (8)/(3) + 14 ) - ( (1)/(3) + 7 ) A = ( (8)/(3) + (42)/(3) ) - ( (1)/(3) + (21)/(3) ) A = (50)/(3) - (22)/(3) = (28)/(3) The area is (28)/(3) square units. 9) Step 1: Set up the definite integral for the area under the curve y = x^2 - 3x + 4 from x=0 to x=4. A = _0^4 (x^2 - 3x + 4)\,dx Step 2: Integrate the function term by term. (x^2 - 3x + 4)\,dx = x^2+12+1 - 3x^1+11+1 + 4x = (x^3)/(3) - (3x^2)/(2) + 4x Step 3: Evaluate the definite integral using the Fundamental Theorem of Calculus. A = [ (x^3)/(3) - (3x^2)/(2) + 4x ]_0^4 = ( (4^3)/(3) - (3(4^2))/(2) + 4(4) ) - ( (0^3)/(3) - (3(0^2))/(2) + 4(0) ) Step 4: Calculate the values. A = ( (64)/(3) - (3(16))/(2) + 16 ) - (0) A = (64)/(3) - (48)/(2) + 16 A = (64)/(3) - 24 + 16 A = (64)/(3) - 8 Step 5: Combine the terms. A = (64)/(3) - (24)/(3) = (40)/(3) The area is (40)/(3) square units. What's next?