Question

Let S ₁​ be the area of the region R₁ bounded by the curve y=x ³

and the curve y=ax ² (0≤a≤1), and let S₂ be the area of the region R ₂bounded by the curve y=x ³ , the curve y=ax²
(0≤a≤1), and the line x=1, where a is the same constant for R ₁and R₂

Answer 1

To find the areas of regions bounded by curves and lines, we need to first find the points of intersection. Then, we can calculate the areas by taking integrals over the appropriate intervals.

Step-by-step explanation:

Solution:

The area of region R₁ bounded by the curve y=x³ and the curve y=ax² (0≤a≤1) can be found by taking the integral of the difference between the two curves over the interval where they intersect. Let's find the points of intersection first:

x³ = ax²

x³ - ax² = 0

x²(x - a) = 0

The solutions are x = 0 and x = a, so these are the bounds of integration for R₁.

Now, we can calculate the area:

S₁ = ∫(ax² - x³)dx from x = 0 to x = a

To find the area of region R₂ bounded by the curve y=x³, the curve y=ax², and the line x=1, we need to find the points of intersection between the curve y=ax² and the line x=1:

ax² = 1

x = √(1/a)

So, the bounds of integration for R₂ are x = 0 and x = √(1/a).

S₂ = ∫(ax² - x³)dx from x = 0 to x = √(1/a)