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.
ScanSolve AI Answer
You're on a roll — Step 1: Set up the integral for the area between the curves. The problem provides the equations for the two curves, y_1 = sqrt(ax) and y_2 = (x^2)/(a), and the integral setup: A = _0^a (y_1 - y_2) \,dx Substitute the expressions for y_1 and y_2: A = _0^a (sqrt(ax) - (x^2)/(a)) \,dx Step 2: Rewrite the terms for easier integration. sqrt(ax) = sqrt(a) · sqrt(x) = sqrt(a) x^1/2 (x^2)/(a) = (1)/(a) x^2 So the integral becomes: A = _0^a (sqrt(a) x^1/2 - (1)/(a) x^2) \,dx Step 3: Integrate each term. Recall the power rule for integration: x^n \,dx = x^n+1n+1 + C. sqrt(a) x^1/2 \,dx = sqrt(a) x^1/2+11/2+1 = sqrt(a) x^3/23/2 = (2)/(3) sqrt(a) x^3/2 (1)/(a) x^2 \,dx = (1)/(a) x^2+12+1 = (1)/(a) (x^3)/(3) = (x^3)/(3a) Step 4: Evaluate the definite integral from 0 to a. A = [ (2)/(3) sqrt(a) x^3/2 - (x^3)/(3a) ]_0^a Substitute the upper limit x=a: ( (2)/(3) sqrt(a) (a)^3/2 - ((a)^3)/(3a) ) Simplify the terms: (2)/(3) a^1/2 a^3/2 - (a^3)/(3a) = (2)/(3) a^(1/2 + 3/2) - (a^2)/(3) = (2)/(3) a^4/2 - (a^2)/(3) = (2)/(3) a^2 - (a^2)/(3) Substitute the lower limit x=0: ( (2)/(3) sqrt(a) (0)^3/2 - ((0)^3)/(3a) ) = 0 - 0 = 0 Subtract the lower limit value from the upper limit value: A = ( (2)/(3) a^2 - (a^2)/(3) ) - 0 A = (2a^2 - a^2)/(3) = (a^2)/(3) The final answer is (a^2)/(3). What's next?