If the area under the curve y= a^2-x^2 included between the lines x=0 and x=a is 4 sq units. Then the value of ' a ' is

Step-by-step solution

The area under the curve y = a^2 - x^2 from x = 0 to x = a is given by the integral A = ₀^ a a^2 - x^2 dx . Using the standard integral formula a^2 - x^2 dx = x 2 a^2 - x^2 + a^2 2 ⁻¹ ( x a ) , we evaluate the definite integral: A = [ x 2 a^2 - x^2 + a^2 2 ⁻¹ ( x a ) ]₀^ a Substituting the limits: A = ( a 2 a^2 - a^2 + a^2 2 ⁻¹ (1) ) - ( 0 + a^2 2 ⁻¹ (0) ) A = 0 + a^2 2 ( 2 ) - 0 = a^2 4 Given that the area A = 4 , we have a^2 4 = 4 . Solving for a^2 : a^2 = 16 . Therefore, a = 16 = 4 . Answer: 4