Final answer:
To find the value of a that minimizes the value of S₁+S₂, we need to first find the points where the two curves intersect. Next, we integrate the functions y = x³ and y = ax² from x = 0 to x = √a to find the areas under the curves. Finally, we can calculate the total area and find the value of a that minimizes it.
Step-by-step explanation:
To find the value of a that minimizes the value of S₁+S₂, we need to first find the points where the two curves intersect. Setting y = x³ equal to y = ax² and solving for x, we get x³ = ax². Rearranging the equation gives us x(x² - a) = 0. This equation has solutions x = 0 and x = √a.
Next, we integrate the functions y = x³ and y = ax² from x = 0 to x = √a to find the areas under the curves. The integral of y = x³ is given by S₁ = ∫(x³)dx = (1/4)x⁴, and the integral of y = ax² is given by S₂ = ∫(ax²)dx = (a/3)x³.
Now, we can substitute the limits of integration into the area formulas to get the areas in terms of a. S₁ = (1/4)(√a)⁴ = a⁴/4 and S₂ = (a/3)(√a)³ = a⁴/3. Finally, we can calculate the total area by summing S₁ and S₂, giving us S = S₁ + S₂ = a⁴/4 + a⁴/3.
To find the value of a that minimizes S, we can take the derivative of S with respect to a and set it equal to zero. Taking the derivative, we get dS/da = (4a³)/4 + (4a³)/3 = (7a³)/3. Setting this equal to zero and solving for a, we find a = 0. Therefore, the value of a that minimizes the value of S is a = 0.