Find the range of the function f of x equals the integral from 0 to x of the square root of the quantity 16 minus t squared

Question

Find the range of the function f of x equals the integral from 0 to x of the square root of the quantity 16 minus t squared

A.[-4, 4]

B.[0, 4]

C.[0, 4π]

D.[0, 8π]

Answer 1

The correct answer is [-4π, 4π], which is none of the provided.

In a coordinate plane where y is the vertical axis and t is the horizontal axis,

the function

y = √(16 - t²)

is a semi-circle of radius 4 above the t-axis.

The domain of the function y(t) is [-4, 4].

Therefore the domain of f(x) is also [-4, 4].

If we restrict the domain of f(x) to [0, 4] then each value of f(x)

is a partial area under the quarter circle.

The maximum it can attain is the whole quarter-circle, whose area is

π4²/4 = 4π.

Therefore if f(x) is restricted to the domain [0,4] then its range is [0, 4π].

If f(x) were allowed its whole domain [-4,4] then its range would be [-4π, 4π],

because f(-x) = f(x).