Let f(x) = 1/√(4-x^2). a.) Determine ∫f(x)dx. b.) What is the area of the region bounded by f, the x-axis, the y-axis, and the line x=1.

Question

Let f(x) = 1/√(4-x^2).

a.) Determine ∫f(x)dx.

b.) What is the area of the region bounded by f, the x-axis, the y-axis, and the line x=1.

c.) Determine the value of a that makes the following equality true: ∫^a₁f(x)dx=∫¹₀f(x)

Please show all work.

Answer 1

a) ∫1/√(4-x^2)dx = sin^-1(x/2) + C

b) Area A = ∫₀¹1/(4 - x²)dx = sin^-1(x/2)₀¹ = sin^-1(1/2) - sin^-1(0) = π/6

c) ∫₀¹1/√(4 - x²)dx=∫₁^a1/√(4 - x²)dx; sin^-1(x/2)₀¹ = sin^-1(x/2)₁^a; sin^-1(1/2) = sin^-1(a/2) - sin^-1(1/2)

sin^-1(a/2) = π/3; a/2 = √3/2; a = √3